Acc Velo Dist
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velocity - Kinematics - Physics Code in C, C++ and Excel
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physics › kinematics ›
velocity
Private Project
velocity - Kinematics - Physics Code in C, C++ and Excel
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#include #include
average and instantaneous velocity of an object int main() { // final position and time double x = 100, t = 15.7;
Contents 1. Interface 2. Function Documentation 3. Page Comments
std::cout std::cout std::cout std::cout
Private project under development, to help contact the author:
Group Description This module computes the average and instantaneous velocity of a moving object at given moments of time.
<< << << <<
std::endl; " Final position = " << x << " m" << std::endl; " Time spent = " << t << " s" << std::endl; std::endl;
// assuming initial position and initial time are null, // display the average velocity of the object
Authors: Lucian Bentea (July 2007)
std::cout << "Average velocity = " << Physics::Kinematics::velocity_avg(x, t); std::cout << " m/s" << std::endl;
Interface
return 0;
#include
} using namespace Physics::Kinematics;
Output double
velocity_avg (double xf, double tf, double x0 = 0, double t0 = 0)
Final position = 100 m Time spent = 15.7 s
[inline] Average velocity of an object given its initial and final position, and also the time spent Real
Average velocity = 6.36943 m/s
cc_velocity_avg (Real xf, Real tf, Real x0, Real t0)
Parameters:
This function is available as a Microsoft Excel add-in. double
xf
final position on the axis (meters)
tf
final time (seconds) [needs to be different from t0]
Instantaneous velocity of an object at a certain moment of time,
x0
Default value = 0
given the position function
t0
Default value = 0
velocity_ins (double (*x)
Returns:
velocity_ins (double (*x)(double), double t, double eps = 1E-6) [inline]
std::vector<double>
(double), std::vector<double> &t, double eps = 1E-6)
[inline]
the average velocity of the moving object (meters per second)
Instantaneous velocities of an object at several moments of time,
double velocity_ins( double (*x)(double) [function pointer]
given the position function
double t double eps = 1E-6
Function Documentation Add calculator to website or email
double velocity_avg( double xf double tf double x0 = 0
) [inline]
This function returns the instantaneous velocity of a moving object at a certain moment of time t, given a function x which determines the position of the object at any moment of time. It is based on the fact that with respect the instantaneous velocity function is given by the derivative of the position function to time, i.e. (2)
double t0 = 0 ) [inline] This function computes the average velocity of a moving object, given its initial and final position on an axis and also the total time spent to get to that final position. Considering is the initial position at time and is the final position at time , the average velocity is given by the following simple formula: (1)
Since this function uses numerical differentiation to compute the above derivative, an optional parameter eps is available to specify the precision of numerical computations. Example 2: #include #include
Example 1:
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http://www.codecogs.com/d-ox/physics/kinematics/velocity.php
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velocity - Kinematics - Physics Code in C, C++ and Excel
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velocity - Kinematics - Physics Code in C, C++ and Excel
// function defining the position at any moment of time t; // in this case pos(t) = t^3
#include // function defining the position at any moment of time t; // in this case pos(t) = t^3
double pos(double t) { return t*t*t; }
double pos(double t) { return t*t*t; }
int main() { // time at which to calculate instantaneous velocity double t = 11.43; std::cout std::cout std::cout std::cout std::cout std::cout
<< << << << << <<
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int main() { // moments of time at which to evaluate // the instantaneous velocity of the object
std::endl; "Position = " << pos(t); " m" << std::endl; " Time = " << t; " s" << std::endl; std::endl;
double t[10] = { 11.40, 11.41, 11.42, 11.43, 11.44, 11.45, 11.46, 11.47, 11.48, 11.49 };
// display instantaneous velocity at time t
// compute the instantaneous velocities
std::cout << "Instantaneous velocity = " << Physics::Kinematics::velocity_ins(space, t); std::cout << " m/s" << std::endl;
std::vector<double> time(t, t+10), velocities = Physics::Kinematics::velocity_ins(distance, time); // display the time, the position // and the instantaneous velocities
return 0; }
std::cout << std::endl; for (int i = 0; i < 10; i++) { std::cout << "Time = " << time[i] << " s"; std::cout << "\tPosition = " << pos(time[i]) << " m"; std::cout << "\tVelocity = " << velocities[i] << " m/s"; std::cout << std::endl; }
Output Position = 1493.27 m Time = 11.43 s Instantaneous velocity = 391.935 m/s Parameters: x
function defining the position of the object at any moment of time (meters)
t
the moment of time at which the instantaneous velocity is to be evaluated (seconds)
eps
Default value = 1E-6
return 0; } Output
Returns: the instantaneous velocity of the object at time t (meters per second) std::vector<double> velocity_ins double (
Time Time Time Time Time Time Time Time Time Time
(*x)(double) [function pointer]
std::vector<double> & t double
eps = 1E-6
) [inline]
This function is based on the same equation as the previous one, only that it is able to compute the instantaneous velocities at several moments of time and return the results in the form of an array. Notice the example code below which shows exactly how this is a generalisation of the previous overloaded function. Example 3:
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11.4 s 11.41 s 11.42 s 11.43 s 11.44 s 11.45 s 11.46 s 11.47 s 11.48 s 11.49 s
Position Position Position Position Position Position Position Position Position Position
= = = = = = = = = =
1481.54 1485.45 1489.36 1493.27 1497.19 1501.12 1505.06 1509 m 1512.95 1516.91
m m m m m m m m m
Velocity Velocity Velocity Velocity Velocity Velocity Velocity Velocity Velocity Velocity
= = = = = = = = = =
389.88 m/s 390.564 m/s 391.249 m/s 391.935 m/s 392.621 m/s 393.307 m/s 393.995 m/s 394.683 m/s 395.371 m/s 396.06 m/s
Parameters: x
function defining the position of the object at any moment of time (meters)
t
array containing the moments of time at which the instantaneous velocities should be evaluated (seconds)
eps
Default value = 1E-6
#include
http://www.codecogs.com/d-ox/physics/kinematics/velocity.php
= = = = = = = = = =
http://www.codecogs.com/d-ox/physics/kinematics/velocity.php
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velocity - Kinematics - Physics Code in C, C++ and Excel
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position uniform - Kinematics - Physics Code in C, C++ and Excel
Page 1 of 3
physics › kinematics ›
Returns: array containing the instantaneous velocities of the object at moments of time given by t (meters per second)
Page Comments
position uniform
Private Project
position of an object moving uniformly Contents 1. Interface 2. Function Documentation 3. Page Comments
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Private project under development, to help contact the author:
Group Description
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This module computes the position of an object moving uniformly (constant speed or constant acceleration). Authors: Lucian Bentea (July 2007)
Interface #include using namespace Physics::Kinematics; double
position_const_velocity (double t, double v, double x0 = 0, double t0 = 0)
[inline]
Position of an object moving at constant velocity Real
cc_position_const_velocity (Real t, Real v, Real x0, Real t0) This function is available as a Microsoft Excel add-in.
double
position_const_acceleration (double t, double a, double v0, double x0 = 0, double t0 = 0) [inline] Position of an object moving at constant acceleration
Real
cc_position_const_acceleration (Real t, Real a, Real v0, Real x0, Real t0) This function is available as a Microsoft Excel add-in.
Function Documentation Add calculator to website or email
double position_const_velocity (
double t double v double
x0 = 0
double t0 = 0 ) [inline] This function determines the position of an object at time after moving uniformly at constant velocity for a certain period of time. The formula relating the above quantities is (1) where
is the initial position of the object at time
.
Example 1: #include #include int main() {
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http://www.codecogs.com/d-ox/physics/kinematics/position_uniform.php
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position uniform - Kinematics - Physics Code in C, C++ and Excel
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position uniform - Kinematics - Physics Code in C, C++ and Excel
// the constant velocity and the current time double v = 5.7, t = 12;
// the constant velocity and the current time double a = 0.7, v0 = 4.33, t = 9.8;
std::cout std::cout std::cout std::cout std::cout
std::cout std::cout std::cout std::cout std::cout std::cout std::cout
<< << << << <<
std::endl; "Velocity = " << v << " m/s"; std::endl; " Time = " << t << " s"; std::endl << std::endl;
// compute the current position // assuming the initial position and time are null
<< << << << << << <<
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std::endl; " Acceleration = " << a << " m/s^2"; std::endl; "Init. velocity = " << v0 << " m/s"; std::endl; " Time = " << t << " s"; std::endl << std::endl;
// compute the current position // assuming the initial position and time are null
std::cout << "Position = " << Physics::Kinematics::position_const_velocity(t, v); std::cout << " m" << std::endl;
std::cout << " Position = " << Physics::Kinematics::position_const_acceleration(t, a, v0); std::cout << " m" << std::endl;
return 0; }
return 0; Output:
}
Velocity = 5.7 m/s Time = 12 s
Output: Acceleration = 0.7 m/s^2 Init. velocity = 4.33 m/s Time = 9.8 s
Position = 68.4 m Parameters: t
the current time (seconds)
v
the value of the constant velocity (meters per second)
x0
Default value = 0
t0
Default value = 0
Position = 76.048 m Parameters:
Returns: the position of the object after moving uniformly at constant velocity for t seconds (meters)
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double position_const_acceleration double t (
t
the current time (seconds)
a
the value of the constant acceleration (meters per sq. second)
v0
the initial velocity of the object at time t0 (meters per second)
x0
Default value = 0
t0
Default value = 0
Returns:
double a
the position of the object after moving uniformly at constant acceleration for t seconds (meters)
double v0 double
x0 = 0
double
t0 = 0
Page Comments ) [inline]
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Last Modified: 18 Oct 07 @ 17:07
This function determines the position of an object at time after moving uniformly at constant acceleration for a certain period of time, with initial velocity given by . The formula relating the above quantities is
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(2) where
is the initial position of the object at time
.
Example 2: #include #include int main() {
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position - Kinematics - Physics Code in C, C++ and Excel
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position - Kinematics - Physics Code in C, C++ and Excel
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physics › kinematics ›
position
std::cout std::cout std::cout std::cout std::cout
Private Project
position of an object moving uniformly Contents 1. Interface 2. Function Documentation 3. Page Comments
<< << << << <<
std::endl; "Velocity = " << v << " m/s"; std::endl; " Time = " << t << " s"; std::endl << std::endl;
// compute the current position // assuming the initial position and time are null
Private project under development, to help contact the author:
std::cout << "Position = " << Physics::Kinematics::position_velocity(t, v); std::cout << " m" << std::endl;
Group Description This module computes the position of an object moving uniformly (constant speed or constant acceleration).
return 0; }
Authors: Lucian Bentea (July 2007)
Output: Velocity = 5.7 m/s Time = 12 s
Interface #include
Position = 68.4 m
using namespace Physics::Kinematics; double
Real
double
Parameters:
position_velocity (double t, double v, double x0 = 0, double t0 = 0)
[inline]
t
the current time (seconds)
Position of an object moving at constant velocity
v
the value of the constant velocity (meters per second)
cc_position_velocity (Real t, Real v, Real x0, Real t0)
x0
Default value = 0
This function is available as a Microsoft Excel add-in.
t0
Default value = 0
position_acceleration (double t, double a, double v0, double x0 = 0, double t0 = 0)
Returns:
[inline]
the position of the object after moving uniformly at constant velocity for t seconds (meters)
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Position of an object moving at constant acceleration Real
double position_acceleration( double t
cc_position_acceleration (Real t, Real a, Real v0, Real x0, Real t0)
double a
This function is available as a Microsoft Excel add-in.
double v0 double x0 = 0
Function Documentation
double t0 = 0 ) [inline] Add calculator to website or email
double position_velocity( double t
This function determines the position of an object at time after moving uniformly at constant acceleration for a certain period of time, with initial velocity given by . The formula relating the above quantities is
double v double x0 = 0
(2)
double t0 = 0 ) [inline] where
This function determines the position of an object at time after moving uniformly at constant velocity for a certain period of time. The formula relating the above quantities is (1) where
is the initial position of the object at time
is the initial position of the object at time
.
Example 2:
.
#include #include
Example 1: int main() { // the constant velocity and the current time double a = 0.7, v0 = 4.33, t = 9.8;
#include #include int main() { // the constant velocity and the current time double v = 5.7, t = 12;
http://www.codecogs.com/d-ox/physics/kinematics/position.php
std::cout << std::endl; std::cout << " Acceleration = " << a << " m/s^2";
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http://www.codecogs.com/d-ox/physics/kinematics/position.php
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position - Kinematics - Physics Code in C, C++ and Excel std::cout std::cout std::cout std::cout std::cout
<< << << << <<
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std::endl; "Init. velocity = " << v0 << " m/s"; std::endl; " Time = " << t << " s"; std::endl << std::endl;
acceleration - Kinematics - Physics Code in C, C++ and Excel
Page 1 of 7
physics › kinematics ›
acceleration
Private Project
average and instantaneous acceleration of an object Contents
// compute the current position // assuming the initial position and time are null
1. Interface 2. Function Documentation 3. Page Comments
Position = " << std::cout << " Physics::Kinematics::position_acceleration(t, a, v0); std::cout << " m" << std::endl;
Private project under development, to help contact the author:
Group Description
return 0;
This module computes the average and instantaneous acceleration of a moving object at given moments of time.
}
Authors:
Output:
Lucian Bentea (July 2007) Acceleration = 0.7 m/s^2 Init. velocity = 4.33 m/s Time = 9.8 s
Interface #include
Position = 76.048 m
using namespace Physics::Kinematics;
Parameters:
double
t
the current time (seconds)
a
the value of the constant acceleration (meters per sq. second)
v0
the initial velocity of the object at time t0 (meters per second)
x0
Default value = 0
t0
Default value = 0
acceleration_avg (double vf, double tf, double v0 = 0, double t0 = 0) [inline] Average acceleration of an object given the initial and final velocities, and also the time spent
Real
cc_acceleration_avg (Real vf, Real tf, Real v0, Real t0) This function is available as a Microsoft Excel add-in.
Returns: the position of the object after moving uniformly at constant acceleration for t seconds (meters)
double
acceleration_ins (double (*v)(double), double t, double eps = 1E-6) [inline] Instantaneous acceleration of an object at a certain moment of time,
Page Comments
given the velocity function std::vector<double>
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acceleration_ins (double (*v) (double), std::vector<double> &t, double eps = 1E-6)
[inline]
Instantaneous acceleration values of an object at several moments of Last Modified: 18 Oct 07 @ 17:07
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time, given the velocity function double
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acceleration_ins_space (double (*x) (double), double t, double eps = 1E-5)
[inline]
Instantaneous acceleration of an object at a certain moment of time, given the position function std::vector<double>
acceleration_ins_space (double (*x) (double), std::vector<double> &t, double eps = 1E-5)
[inline]
Instantaneous acceleration values of an object at several moments of time, given the position function
Function Documentation Add calculator to website or email
double acceleration_avg( double vf double tf double v0 = 0 double t0 = 0 ) [inline]
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acceleration - Kinematics - Physics Code in C, C++ and Excel
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This function computes the average acceleration of a moving object, given the initial and final velocities, and also the total time spent. Considering is the initial velocity at time and is the final velocity at time , the average acceleration is given by the following simple formula: (1)
acceleration - Kinematics - Physics Code in C, C++ and Excel
Page 3 of 7
Since this function uses numerical differentiation to compute the above derivative, an optional parameter eps is available to specify the precision of numerical computations. Example 2: #include #include
Example 1: #include #include
// function defining the velocity at any moment of time t; // in this case velocity(t) = t^2/2
int main() { // final velocity and time double v = 200, t = 15.7; std::cout std::cout std::cout std::cout
<< << << <<
double velocity(double t) { return t*t/2; }
std::endl; " Final velocity = " << v << " m/s" << std::endl; " Time spent = " << t << " s" << std::endl; std::endl;
int main() { // time at which to calculate instantaneous acceleration double t = 11.43;
// assuming initial velocity and initial time are null, // display the average acceleration of the object
std::cout std::cout std::cout std::cout std::cout std::cout
std::cout << "Average acceleration = " << Physics::Kinematics::acceleration_avg(v, t); std::cout << " m/s^2" << std::endl;
<< << << << << <<
std::endl; "Velocity = " << velocity(t); " m/s" << std::endl; " Time = " << t; " s" << std::endl; std::endl;
return 0; // display instantaneous acceleration at time t
} Output
std::cout << "Instantaneous acceleration = " << Physics::Kinematics::acceleration_ins(velocity, t); std::cout << " m/s^2" << std::endl;
Final velocity = 200 m/s Time spent = 15.7 s
return 0;
Average acceleration = 12.7389 m/s^2 } Parameters: vf
final velocity (meters per second)
tf
final time (seconds) [needs to be different from t0]
v0
Default value = 0
t0
Default value = 0
Output Velocity = 65.3225 m/s Time = 11.43 s Instantaneous acceleration = 11.43 m/s^2
Returns: Parameters:
the average acceleration of the moving object (meters per sq. second) double acceleration_ins( double (*v)(double) [function pointer] double t double eps = 1E-6
) [inline]
v
function defining the velocity of the object at any moment of time (meters per second)
t
the moment of time at which the instantaneous acceleration is to be evaluated (seconds)
eps
Default value = 1E-6
Returns: the instantaneous acceleration of the object at time t (meters per sq. second)
This function returns the instantaneous acceleration of a moving object at a certain moment of time t, given a function v which describes the velocity of the object at any moment of time. It is based on the fact that the instantaneous acceleration function is given by the derivative of the velocity function with respect to time, i.e. (2)
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std::vector<double> acceleration_ins double (
(*v)(double) [function pointer]
std::vector<double> & t double
eps = 1E-6
http://www.codecogs.com/d-ox/physics/kinematics/acceleration.php
)
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acceleration - Kinematics - Physics Code in C, C++ and Excel
Page 4 of 7
acceleration - Kinematics - Physics Code in C, C++ and Excel Time Time Time Time Time
[inline] This function is based on the same equation as the previous one, only that it is able to compute the instantaneous acceleration values at several moments of time and return the results in the form of an array. Notice the example code below which shows exactly how this is a generalisation of the previous overloaded function. Example 3: #include #include
= = = = =
11.45 11.46 11.47 11.48 11.49
s s s s s
Velocity Velocity Velocity Velocity Velocity
= = = = =
65.5512 65.6658 65.7805 65.8952 66.0101
m/s m/s m/s m/s m/s
Acceleration Acceleration Acceleration Acceleration Acceleration
= = = = =
11.45 11.46 11.47 11.48 11.49
Page 5 of 7 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2
Parameters: v
function defining the velocity at any moment of time (meters per second)
t
array containing the moments of time at which the instantaneous acceleration should be evaluated (seconds)
eps
Default value = 1E-6
Returns: array containing the instantaneous acceleration values at the moments of time given by t (meters per sq. second)
// function defining the velocity at any moment of time t; // in this case velocity(t) = t^2/2
double acceleration_ins_space( double (*x)(double) [function pointer] double speed(double t) { return t*t/2; }
double t double eps = 1E-5
This function returns the instantaneous acceleration of a moving object at a certain moment of time t, given a function x which determines the position of the object at any moment of time on a fixed axis. It is based on the fact that the instantaneous acceleration function is given by the second order derivative of the position function with respect to time, i.e.
int main() { // moments of time at which to evaluate // the instantaneous acceleration of the object
(3)
double t[10] = { 11.40, 11.41, 11.42, 11.43, 11.44, 11.45, 11.46, 11.47, 11.48, 11.49 };
Since this function uses numerical differentiation to compute the above second order derivative, an optional parameter eps is available to specify the precision of numerical computations. Example 4: #include #include
// compute the instantaneous acceleration values std::vector<double> time(t, t+10), acceleration = Physics::Kinematics::acceleration_ins(speed, time);
// function defining the position at any moment of time t; // in this case space(t) = t^3
// display the time, the velocity // and the instantaneous acceleration
double pos(double t) { return t*t*t; }
std::cout << std::endl; for (int i = 0; i < 10; i++) { std::cout << "Time = " << time[i] << " s"; std::cout << "\tVelocity = " << speed(time[i]) << " m/s"; std::cout << "\tAcceleration = " << acceleration[i] << " m/s^2"; std::cout << std::endl; }
int main() { // time at which to calculate instantaneous acceleration double t = 11.43; std::cout std::cout std::cout std::cout std::cout std::cout
return 0; } Output Time Time Time Time Time
= = = = =
11.4 s 11.41 s 11.42 s 11.43 s 11.44 s
Velocity Velocity Velocity Velocity Velocity
= = = = =
64.98 m/s 65.094 m/s 65.2082 m/s 65.3225 m/s 65.4368 m/s
) [inline]
Acceleration Acceleration Acceleration Acceleration Acceleration
= = = = =
11.4 m/s^2 11.41 m/s^2 11.42 m/s^2 11.43 m/s^2 11.44 m/s^2
http://www.codecogs.com/d-ox/physics/kinematics/acceleration.php
<< << << << << <<
std::endl; "Position = " << space(t); " m" << std::endl; " Time = " << t; " s" << std::endl; std::endl;
// display instantaneous acceleration at time t std::cout << "Instantaneous acceleration = " <<
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http://www.codecogs.com/d-ox/physics/kinematics/acceleration.php
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acceleration - Kinematics - Physics Code in C, C++ and Excel
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acceleration - Kinematics - Physics Code in C, C++ and Excel
Physics::Kinematics::acceleration_ins_space(space, t); std::cout << " m/s^2" << std::endl;
Page 7 of 7
std::vector<double> time(t, t+10), acceleration = Physics::Kinematics::acceleration_ins_space(distance, time);
return 0; }
// display the time, the position // and the instantaneous acceleration values
Output std::cout << std::endl; for (int i = 0; i < 10; i++) { std::cout << "Time = " << time[i] << " s"; std::cout << "\tPosition = " << distance(time[i]) << " m"; std::cout << "\tAcceleration = " << acceleration[i] << " m/s^2"; std::cout << std::endl; }
Position = 1493.27 m Time = 11.43 s Instantaneous acceleration = 68.5782 m/s^2 Parameters: x
function defining the position of the object at any moment of time (meters)
t
the moment of time at which the instantaneous acceleration is to be evaluated (seconds)
eps
Default value = 1E-5 return 0;
Returns:
}
the instantaneous acceleration of the object at time t (meters per sq. second) Output std::vector<double> acceleration_ins_space double (
(*x)(double) [function pointer]
Time Time Time Time Time Time Time Time Time Time
std::vector<double> & t double
eps = 1E-5
) [inline]
This function is based on the same equation as the previous one, only that it is able to compute the instantaneous acceleration values at several moments of time and return the results in the form of an array. Notice the example code below which shows how this is a generalisation of the previous overloaded function. Example 5: #include #include
= = = = = = = = = =
11.4 s 11.41 s 11.42 s 11.43 s 11.44 s 11.45 s 11.46 s 11.47 s 11.48 s 11.49 s
Position Position Position Position Position Position Position Position Position Position
= = = = = = = = = =
1481.54 1485.45 1489.36 1493.27 1497.19 1501.12 1505.06 1509 m 1512.95 1516.91
m m m m m m m m m
Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration
= = = = = = = = = =
68.3985 68.4599 68.5168 68.5782 68.6396 68.6987 68.7601 68.8237 68.8829 68.9397
m/s^2 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2
Parameters: x
function defining the position at any moment of time (meters)
t
array containing the moments of time at which the instantaneous acceleration values should be evaluated (seconds)
eps
Default value = 1E-5
Returns: // function defining the position at any moment of time t; // in this case pos(t) = t^3
array containing the instantaneous acceleration values of the object at moments of time given by t (meters per sq. second)
double pos(double t) { return t*t*t; }
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int main() { // moments of time at which to evaluate // the instantaneous acceleration of the object
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double t[10] = { 11.40, 11.41, 11.42, 11.43, 11.44, 11.45, 11.46, 11.47, 11.48, 11.49 }; // compute the instantaneous acceleration values
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physics › kinematics ›
velocity
Private Project
velocity - Kinematics - Physics Code in C, C++ and Excel
Page 2 of 5
#include
average and instantaneous velocity of an object int main() { // final position and time double x = 100, t = 15.7;
Contents 1. Interface 2. Function Documentation 3. Page Comments
std::cout std::cout std::cout std::cout
Private project under development, to help contact the author:
Group Description This module computes the average and instantaneous velocity of a moving object at given moments of time.
<< << << <<
std::endl; " Final position = " << x << " m" << std::endl; " Time spent = " << t << " s" << std::endl; std::endl;
// assuming initial position and initial time are null, // display the average velocity of the object
Authors: Lucian Bentea (July 2007)
std::cout << "Average velocity = " << Physics::Kinematics::velocity_avg(x, t); std::cout << " m/s" << std::endl;
Interface
return 0;
#include
} using namespace Physics::Kinematics;
Output double
velocity_avg (double xf, double tf, double x0 = 0, double t0 = 0)
Final position = 100 m Time spent = 15.7 s
[inline] Average velocity of an object given its initial and final position, and also the time spent Real
Average velocity = 6.36943 m/s
cc_velocity_avg (Real xf, Real tf, Real x0, Real t0)
Parameters:
This function is available as a Microsoft Excel add-in. double
xf
final position on the axis (meters)
tf
final time (seconds) [needs to be different from t0]
Instantaneous velocity of an object at a certain moment of time,
x0
Default value = 0
given the position function
t0
Default value = 0
velocity_ins (double (*x)
Returns:
velocity_ins (double (*x)(double), double t, double eps = 1E-6) [inline]
std::vector<double>
(double), std::vector<double> &t, double eps = 1E-6)
[inline]
the average velocity of the moving object (meters per second)
Instantaneous velocities of an object at several moments of time,
double velocity_ins( double (*x)(double) [function pointer]
given the position function
double t double eps = 1E-6
Function Documentation Add calculator to website or email
double velocity_avg( double xf double tf double x0 = 0
) [inline]
This function returns the instantaneous velocity of a moving object at a certain moment of time t, given a function x which determines the position of the object at any moment of time. It is based on the fact that with respect the instantaneous velocity function is given by the derivative of the position function to time, i.e. (2)
double t0 = 0 ) [inline] This function computes the average velocity of a moving object, given its initial and final position on an axis and also the total time spent to get to that final position. Considering is the initial position at time and is the final position at time , the average velocity is given by the following simple formula: (1)
Since this function uses numerical differentiation to compute the above derivative, an optional parameter eps is available to specify the precision of numerical computations. Example 2: #include
Example 1:
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http://www.codecogs.com/d-ox/physics/kinematics/velocity.php
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velocity - Kinematics - Physics Code in C, C++ and Excel
Page 3 of 5
velocity - Kinematics - Physics Code in C, C++ and Excel
// function defining the position at any moment of time t; // in this case pos(t) = t^3
#include
double pos(double t) { return t*t*t; }
double pos(double t) { return t*t*t; }
int main() { // time at which to calculate instantaneous velocity double t = 11.43; std::cout std::cout std::cout std::cout std::cout std::cout
<< << << << << <<
Page 4 of 5
int main() { // moments of time at which to evaluate // the instantaneous velocity of the object
std::endl; "Position = " << pos(t); " m" << std::endl; " Time = " << t; " s" << std::endl; std::endl;
double t[10] = { 11.40, 11.41, 11.42, 11.43, 11.44, 11.45, 11.46, 11.47, 11.48, 11.49 };
// display instantaneous velocity at time t
// compute the instantaneous velocities
std::cout << "Instantaneous velocity = " << Physics::Kinematics::velocity_ins(space, t); std::cout << " m/s" << std::endl;
std::vector<double> time(t, t+10), velocities = Physics::Kinematics::velocity_ins(distance, time); // display the time, the position // and the instantaneous velocities
return 0; }
std::cout << std::endl; for (int i = 0; i < 10; i++) { std::cout << "Time = " << time[i] << " s"; std::cout << "\tPosition = " << pos(time[i]) << " m"; std::cout << "\tVelocity = " << velocities[i] << " m/s"; std::cout << std::endl; }
Output Position = 1493.27 m Time = 11.43 s Instantaneous velocity = 391.935 m/s Parameters: x
function defining the position of the object at any moment of time (meters)
t
the moment of time at which the instantaneous velocity is to be evaluated (seconds)
eps
Default value = 1E-6
return 0; } Output
Returns: the instantaneous velocity of the object at time t (meters per second) std::vector<double> velocity_ins double (
Time Time Time Time Time Time Time Time Time Time
(*x)(double) [function pointer]
std::vector<double> & t double
eps = 1E-6
) [inline]
This function is based on the same equation as the previous one, only that it is able to compute the instantaneous velocities at several moments of time and return the results in the form of an array. Notice the example code below which shows exactly how this is a generalisation of the previous overloaded function. Example 3:
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11.4 s 11.41 s 11.42 s 11.43 s 11.44 s 11.45 s 11.46 s 11.47 s 11.48 s 11.49 s
Position Position Position Position Position Position Position Position Position Position
= = = = = = = = = =
1481.54 1485.45 1489.36 1493.27 1497.19 1501.12 1505.06 1509 m 1512.95 1516.91
m m m m m m m m m
Velocity Velocity Velocity Velocity Velocity Velocity Velocity Velocity Velocity Velocity
= = = = = = = = = =
389.88 m/s 390.564 m/s 391.249 m/s 391.935 m/s 392.621 m/s 393.307 m/s 393.995 m/s 394.683 m/s 395.371 m/s 396.06 m/s
Parameters: x
function defining the position of the object at any moment of time (meters)
t
array containing the moments of time at which the instantaneous velocities should be evaluated (seconds)
eps
Default value = 1E-6
#include
http://www.codecogs.com/d-ox/physics/kinematics/velocity.php
= = = = = = = = = =
http://www.codecogs.com/d-ox/physics/kinematics/velocity.php
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velocity - Kinematics - Physics Code in C, C++ and Excel
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position uniform - Kinematics - Physics Code in C, C++ and Excel
Page 1 of 3
physics › kinematics ›
Returns: array containing the instantaneous velocities of the object at moments of time given by t (meters per second)
Page Comments
position uniform
Private Project
position of an object moving uniformly Contents 1. Interface 2. Function Documentation 3. Page Comments
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Private project under development, to help contact the author:
Group Description
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This module computes the position of an object moving uniformly (constant speed or constant acceleration). Authors: Lucian Bentea (July 2007)
Interface #include
position_const_velocity (double t, double v, double x0 = 0, double t0 = 0)
[inline]
Position of an object moving at constant velocity Real
cc_position_const_velocity (Real t, Real v, Real x0, Real t0) This function is available as a Microsoft Excel add-in.
double
position_const_acceleration (double t, double a, double v0, double x0 = 0, double t0 = 0) [inline] Position of an object moving at constant acceleration
Real
cc_position_const_acceleration (Real t, Real a, Real v0, Real x0, Real t0) This function is available as a Microsoft Excel add-in.
Function Documentation Add calculator to website or email
double position_const_velocity (
double t double v double
x0 = 0
double t0 = 0 ) [inline] This function determines the position of an object at time after moving uniformly at constant velocity for a certain period of time. The formula relating the above quantities is (1) where
is the initial position of the object at time
.
Example 1: #include
http://www.codecogs.com/d-ox/physics/kinematics/velocity.php
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position uniform - Kinematics - Physics Code in C, C++ and Excel
Page 2 of 3
position uniform - Kinematics - Physics Code in C, C++ and Excel
// the constant velocity and the current time double v = 5.7, t = 12;
// the constant velocity and the current time double a = 0.7, v0 = 4.33, t = 9.8;
std::cout std::cout std::cout std::cout std::cout
std::cout std::cout std::cout std::cout std::cout std::cout std::cout
<< << << << <<
std::endl; "Velocity = " << v << " m/s"; std::endl; " Time = " << t << " s"; std::endl << std::endl;
// compute the current position // assuming the initial position and time are null
<< << << << << << <<
Page 3 of 3
std::endl; " Acceleration = " << a << " m/s^2"; std::endl; "Init. velocity = " << v0 << " m/s"; std::endl; " Time = " << t << " s"; std::endl << std::endl;
// compute the current position // assuming the initial position and time are null
std::cout << "Position = " << Physics::Kinematics::position_const_velocity(t, v); std::cout << " m" << std::endl;
std::cout << " Position = " << Physics::Kinematics::position_const_acceleration(t, a, v0); std::cout << " m" << std::endl;
return 0; }
return 0; Output:
}
Velocity = 5.7 m/s Time = 12 s
Output: Acceleration = 0.7 m/s^2 Init. velocity = 4.33 m/s Time = 9.8 s
Position = 68.4 m Parameters: t
the current time (seconds)
v
the value of the constant velocity (meters per second)
x0
Default value = 0
t0
Default value = 0
Position = 76.048 m Parameters:
Returns: the position of the object after moving uniformly at constant velocity for t seconds (meters)
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double position_const_acceleration double t (
t
the current time (seconds)
a
the value of the constant acceleration (meters per sq. second)
v0
the initial velocity of the object at time t0 (meters per second)
x0
Default value = 0
t0
Default value = 0
Returns:
double a
the position of the object after moving uniformly at constant acceleration for t seconds (meters)
double v0 double
x0 = 0
double
t0 = 0
Page Comments ) [inline]
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Last Modified: 18 Oct 07 @ 17:07
This function determines the position of an object at time after moving uniformly at constant acceleration for a certain period of time, with initial velocity given by . The formula relating the above quantities is
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(2) where
is the initial position of the object at time
.
Example 2: #include
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http://www.codecogs.com/d-ox/physics/kinematics/position_uniform.php
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position - Kinematics - Physics Code in C, C++ and Excel
Page 1 of 3
position - Kinematics - Physics Code in C, C++ and Excel
Page 2 of 3
physics › kinematics ›
position
std::cout std::cout std::cout std::cout std::cout
Private Project
position of an object moving uniformly Contents 1. Interface 2. Function Documentation 3. Page Comments
<< << << << <<
std::endl; "Velocity = " << v << " m/s"; std::endl; " Time = " << t << " s"; std::endl << std::endl;
// compute the current position // assuming the initial position and time are null
Private project under development, to help contact the author:
std::cout << "Position = " << Physics::Kinematics::position_velocity(t, v); std::cout << " m" << std::endl;
Group Description This module computes the position of an object moving uniformly (constant speed or constant acceleration).
return 0; }
Authors: Lucian Bentea (July 2007)
Output: Velocity = 5.7 m/s Time = 12 s
Interface #include
Position = 68.4 m
using namespace Physics::Kinematics; double
Real
double
Parameters:
position_velocity (double t, double v, double x0 = 0, double t0 = 0)
[inline]
t
the current time (seconds)
Position of an object moving at constant velocity
v
the value of the constant velocity (meters per second)
cc_position_velocity (Real t, Real v, Real x0, Real t0)
x0
Default value = 0
This function is available as a Microsoft Excel add-in.
t0
Default value = 0
position_acceleration (double t, double a, double v0, double x0 = 0, double t0 = 0)
Returns:
[inline]
the position of the object after moving uniformly at constant velocity for t seconds (meters)
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Position of an object moving at constant acceleration Real
double position_acceleration( double t
cc_position_acceleration (Real t, Real a, Real v0, Real x0, Real t0)
double a
This function is available as a Microsoft Excel add-in.
double v0 double x0 = 0
Function Documentation
double t0 = 0 ) [inline] Add calculator to website or email
double position_velocity( double t
This function determines the position of an object at time after moving uniformly at constant acceleration for a certain period of time, with initial velocity given by . The formula relating the above quantities is
double v double x0 = 0
(2)
double t0 = 0 ) [inline] where
This function determines the position of an object at time after moving uniformly at constant velocity for a certain period of time. The formula relating the above quantities is (1) where
is the initial position of the object at time
is the initial position of the object at time
.
Example 2:
.
#include
Example 1: int main() { // the constant velocity and the current time double a = 0.7, v0 = 4.33, t = 9.8;
#include
http://www.codecogs.com/d-ox/physics/kinematics/position.php
std::cout << std::endl; std::cout << " Acceleration = " << a << " m/s^2";
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http://www.codecogs.com/d-ox/physics/kinematics/position.php
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position - Kinematics - Physics Code in C, C++ and Excel std::cout std::cout std::cout std::cout std::cout
<< << << << <<
Page 3 of 3
std::endl; "Init. velocity = " << v0 << " m/s"; std::endl; " Time = " << t << " s"; std::endl << std::endl;
acceleration - Kinematics - Physics Code in C, C++ and Excel
Page 1 of 7
physics › kinematics ›
acceleration
Private Project
average and instantaneous acceleration of an object Contents
// compute the current position // assuming the initial position and time are null
1. Interface 2. Function Documentation 3. Page Comments
Position = " << std::cout << " Physics::Kinematics::position_acceleration(t, a, v0); std::cout << " m" << std::endl;
Private project under development, to help contact the author:
Group Description
return 0;
This module computes the average and instantaneous acceleration of a moving object at given moments of time.
}
Authors:
Output:
Lucian Bentea (July 2007) Acceleration = 0.7 m/s^2 Init. velocity = 4.33 m/s Time = 9.8 s
Interface #include
Position = 76.048 m
using namespace Physics::Kinematics;
Parameters:
double
t
the current time (seconds)
a
the value of the constant acceleration (meters per sq. second)
v0
the initial velocity of the object at time t0 (meters per second)
x0
Default value = 0
t0
Default value = 0
acceleration_avg (double vf, double tf, double v0 = 0, double t0 = 0) [inline] Average acceleration of an object given the initial and final velocities, and also the time spent
Real
cc_acceleration_avg (Real vf, Real tf, Real v0, Real t0) This function is available as a Microsoft Excel add-in.
Returns: the position of the object after moving uniformly at constant acceleration for t seconds (meters)
double
acceleration_ins (double (*v)(double), double t, double eps = 1E-6) [inline] Instantaneous acceleration of an object at a certain moment of time,
Page Comments
given the velocity function std::vector<double>
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acceleration_ins (double (*v) (double), std::vector<double> &t, double eps = 1E-6)
[inline]
Instantaneous acceleration values of an object at several moments of Last Modified: 18 Oct 07 @ 17:07
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time, given the velocity function double
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acceleration_ins_space (double (*x) (double), double t, double eps = 1E-5)
[inline]
Instantaneous acceleration of an object at a certain moment of time, given the position function std::vector<double>
acceleration_ins_space (double (*x) (double), std::vector<double> &t, double eps = 1E-5)
[inline]
Instantaneous acceleration values of an object at several moments of time, given the position function
Function Documentation Add calculator to website or email
double acceleration_avg( double vf double tf double v0 = 0 double t0 = 0 ) [inline]
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acceleration - Kinematics - Physics Code in C, C++ and Excel
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This function computes the average acceleration of a moving object, given the initial and final velocities, and also the total time spent. Considering is the initial velocity at time and is the final velocity at time , the average acceleration is given by the following simple formula: (1)
acceleration - Kinematics - Physics Code in C, C++ and Excel
Page 3 of 7
Since this function uses numerical differentiation to compute the above derivative, an optional parameter eps is available to specify the precision of numerical computations. Example 2: #include
Example 1: #include
// function defining the velocity at any moment of time t; // in this case velocity(t) = t^2/2
int main() { // final velocity and time double v = 200, t = 15.7; std::cout std::cout std::cout std::cout
<< << << <<
double velocity(double t) { return t*t/2; }
std::endl; " Final velocity = " << v << " m/s" << std::endl; " Time spent = " << t << " s" << std::endl; std::endl;
int main() { // time at which to calculate instantaneous acceleration double t = 11.43;
// assuming initial velocity and initial time are null, // display the average acceleration of the object
std::cout std::cout std::cout std::cout std::cout std::cout
std::cout << "Average acceleration = " << Physics::Kinematics::acceleration_avg(v, t); std::cout << " m/s^2" << std::endl;
<< << << << << <<
std::endl; "Velocity = " << velocity(t); " m/s" << std::endl; " Time = " << t; " s" << std::endl; std::endl;
return 0; // display instantaneous acceleration at time t
} Output
std::cout << "Instantaneous acceleration = " << Physics::Kinematics::acceleration_ins(velocity, t); std::cout << " m/s^2" << std::endl;
Final velocity = 200 m/s Time spent = 15.7 s
return 0;
Average acceleration = 12.7389 m/s^2 } Parameters: vf
final velocity (meters per second)
tf
final time (seconds) [needs to be different from t0]
v0
Default value = 0
t0
Default value = 0
Output Velocity = 65.3225 m/s Time = 11.43 s Instantaneous acceleration = 11.43 m/s^2
Returns: Parameters:
the average acceleration of the moving object (meters per sq. second) double acceleration_ins( double (*v)(double) [function pointer] double t double eps = 1E-6
) [inline]
v
function defining the velocity of the object at any moment of time (meters per second)
t
the moment of time at which the instantaneous acceleration is to be evaluated (seconds)
eps
Default value = 1E-6
Returns: the instantaneous acceleration of the object at time t (meters per sq. second)
This function returns the instantaneous acceleration of a moving object at a certain moment of time t, given a function v which describes the velocity of the object at any moment of time. It is based on the fact that the instantaneous acceleration function is given by the derivative of the velocity function with respect to time, i.e. (2)
http://www.codecogs.com/d-ox/physics/kinematics/acceleration.php
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std::vector<double> acceleration_ins double (
(*v)(double) [function pointer]
std::vector<double> & t double
eps = 1E-6
http://www.codecogs.com/d-ox/physics/kinematics/acceleration.php
)
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acceleration - Kinematics - Physics Code in C, C++ and Excel
Page 4 of 7
acceleration - Kinematics - Physics Code in C, C++ and Excel Time Time Time Time Time
[inline] This function is based on the same equation as the previous one, only that it is able to compute the instantaneous acceleration values at several moments of time and return the results in the form of an array. Notice the example code below which shows exactly how this is a generalisation of the previous overloaded function. Example 3: #include
= = = = =
11.45 11.46 11.47 11.48 11.49
s s s s s
Velocity Velocity Velocity Velocity Velocity
= = = = =
65.5512 65.6658 65.7805 65.8952 66.0101
m/s m/s m/s m/s m/s
Acceleration Acceleration Acceleration Acceleration Acceleration
= = = = =
11.45 11.46 11.47 11.48 11.49
Page 5 of 7 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2
Parameters: v
function defining the velocity at any moment of time (meters per second)
t
array containing the moments of time at which the instantaneous acceleration should be evaluated (seconds)
eps
Default value = 1E-6
Returns: array containing the instantaneous acceleration values at the moments of time given by t (meters per sq. second)
// function defining the velocity at any moment of time t; // in this case velocity(t) = t^2/2
double acceleration_ins_space( double (*x)(double) [function pointer] double speed(double t) { return t*t/2; }
double t double eps = 1E-5
This function returns the instantaneous acceleration of a moving object at a certain moment of time t, given a function x which determines the position of the object at any moment of time on a fixed axis. It is based on the fact that the instantaneous acceleration function is given by the second order derivative of the position function with respect to time, i.e.
int main() { // moments of time at which to evaluate // the instantaneous acceleration of the object
(3)
double t[10] = { 11.40, 11.41, 11.42, 11.43, 11.44, 11.45, 11.46, 11.47, 11.48, 11.49 };
Since this function uses numerical differentiation to compute the above second order derivative, an optional parameter eps is available to specify the precision of numerical computations. Example 4: #include
// compute the instantaneous acceleration values std::vector<double> time(t, t+10), acceleration = Physics::Kinematics::acceleration_ins(speed, time);
// function defining the position at any moment of time t; // in this case space(t) = t^3
// display the time, the velocity // and the instantaneous acceleration
double pos(double t) { return t*t*t; }
std::cout << std::endl; for (int i = 0; i < 10; i++) { std::cout << "Time = " << time[i] << " s"; std::cout << "\tVelocity = " << speed(time[i]) << " m/s"; std::cout << "\tAcceleration = " << acceleration[i] << " m/s^2"; std::cout << std::endl; }
int main() { // time at which to calculate instantaneous acceleration double t = 11.43; std::cout std::cout std::cout std::cout std::cout std::cout
return 0; } Output Time Time Time Time Time
= = = = =
11.4 s 11.41 s 11.42 s 11.43 s 11.44 s
Velocity Velocity Velocity Velocity Velocity
= = = = =
64.98 m/s 65.094 m/s 65.2082 m/s 65.3225 m/s 65.4368 m/s
) [inline]
Acceleration Acceleration Acceleration Acceleration Acceleration
= = = = =
11.4 m/s^2 11.41 m/s^2 11.42 m/s^2 11.43 m/s^2 11.44 m/s^2
http://www.codecogs.com/d-ox/physics/kinematics/acceleration.php
<< << << << << <<
std::endl; "Position = " << space(t); " m" << std::endl; " Time = " << t; " s" << std::endl; std::endl;
// display instantaneous acceleration at time t std::cout << "Instantaneous acceleration = " <<
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acceleration - Kinematics - Physics Code in C, C++ and Excel
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acceleration - Kinematics - Physics Code in C, C++ and Excel
Physics::Kinematics::acceleration_ins_space(space, t); std::cout << " m/s^2" << std::endl;
Page 7 of 7
std::vector<double> time(t, t+10), acceleration = Physics::Kinematics::acceleration_ins_space(distance, time);
return 0; }
// display the time, the position // and the instantaneous acceleration values
Output std::cout << std::endl; for (int i = 0; i < 10; i++) { std::cout << "Time = " << time[i] << " s"; std::cout << "\tPosition = " << distance(time[i]) << " m"; std::cout << "\tAcceleration = " << acceleration[i] << " m/s^2"; std::cout << std::endl; }
Position = 1493.27 m Time = 11.43 s Instantaneous acceleration = 68.5782 m/s^2 Parameters: x
function defining the position of the object at any moment of time (meters)
t
the moment of time at which the instantaneous acceleration is to be evaluated (seconds)
eps
Default value = 1E-5 return 0;
Returns:
}
the instantaneous acceleration of the object at time t (meters per sq. second) Output std::vector<double> acceleration_ins_space double (
(*x)(double) [function pointer]
Time Time Time Time Time Time Time Time Time Time
std::vector<double> & t double
eps = 1E-5
) [inline]
This function is based on the same equation as the previous one, only that it is able to compute the instantaneous acceleration values at several moments of time and return the results in the form of an array. Notice the example code below which shows how this is a generalisation of the previous overloaded function. Example 5: #include
= = = = = = = = = =
11.4 s 11.41 s 11.42 s 11.43 s 11.44 s 11.45 s 11.46 s 11.47 s 11.48 s 11.49 s
Position Position Position Position Position Position Position Position Position Position
= = = = = = = = = =
1481.54 1485.45 1489.36 1493.27 1497.19 1501.12 1505.06 1509 m 1512.95 1516.91
m m m m m m m m m
Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration Acceleration
= = = = = = = = = =
68.3985 68.4599 68.5168 68.5782 68.6396 68.6987 68.7601 68.8237 68.8829 68.9397
m/s^2 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2 m/s^2
Parameters: x
function defining the position at any moment of time (meters)
t
array containing the moments of time at which the instantaneous acceleration values should be evaluated (seconds)
eps
Default value = 1E-5
Returns: // function defining the position at any moment of time t; // in this case pos(t) = t^3
array containing the instantaneous acceleration values of the object at moments of time given by t (meters per sq. second)
double pos(double t) { return t*t*t; }
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int main() { // moments of time at which to evaluate // the instantaneous acceleration of the object
Last Modified: 5 Nov 07 @ 14:37
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double t[10] = { 11.40, 11.41, 11.42, 11.43, 11.44, 11.45, 11.46, 11.47, 11.48, 11.49 }; // compute the instantaneous acceleration values
http://www.codecogs.com/d-ox/physics/kinematics/acceleration.php
15/6/2552
http://www.codecogs.com/d-ox/physics/kinematics/acceleration.php
15/6/2552
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