Uses Of Graphs By Sanjay Pandey

Uses of Graphs (i)

(ii)

(iii) (iv)

Sanjay Pandey

[email protected]

The shape of the graph gives a clear idea of the nature of variation of one quantity with respect to the other i.e., it provides us the relationship between the two quantities represented by it. For example, if the graph is a straight line and passes through the origin, it is of the form 𝑦𝑦 = 𝑚𝑚𝑚𝑚 or if the graph is a straight line having positive or negative intercepts on y-axis, it is represented by, 𝑦𝑦 = 𝑚𝑚𝑚𝑚 ± 𝑐𝑐. From the slope of the graph and its intercepts on the x or y-axis, we can calculate the values of physical quantities or universal constants. For example, In photoelectric effect experiment, 𝐸𝐸𝑘𝑘 versus υ graph comes out to be a straight line, giving an intercept on x-axis as well as y-axis. Slope of this straight line graph provides the value of Planck’s constant while the intercepts on y and x-axes give the values of work function and threshold frequency respectively. Inaccuracies in the experimental data can also be identified with the help of a graph. With the help of a graph, we can also find out the mean value from a large number of observations.

Some Common Graphs (i) Straight line graph

(a) Passing through origin : This graph is represented by the equation 𝑦𝑦 = 𝑚𝑚𝑚𝑚. Graph of this nature is obtained when we plot two quantities which are proportional to each other, e.g. velocity-time graph of a body starting from rest and having a uniform acceleration (Fig.1). (b) Not passing through origin : This graph is represented by the equation 𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑐𝑐. Such type of graph is obtained when we plot two quantities which are proportional to each other, but one is not zero when the other is, e.g. velocity-time graph of a body having initial velocity and uniform acceleration (Fig. 2). (c) Having negative slope : Such graph points downwards with the line making an obtuse angle with the x-axis, e.g. velocity-time graph of a body projected vertically upwards till it reaches the maximum height (Fig. 3.)

(ii) Parabola

a) Symmetric parabola : This graph is represented by the equation 𝑦𝑦 = 𝑎𝑎𝑥𝑥 2 . Such type of graph is obtained when one quantity varies directly as the square of the other. The graph passes through the origin and is bisected by one of the axes equally into two symmetrical portions e.g. Kinetic energy and momentum of a body (𝑝𝑝2 = 2𝑚𝑚𝑚𝑚) (Fig. 4) b) Asymmetric parabola : This graph is represented by the equation 𝑦𝑦 = 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑥𝑥 2 . This also passes through the origin and only one portion of graph is obtained at a time, e.g. distance-time graph of body starting with an initial velocity and traveling with uniform acceleration, equation : distance 𝑠𝑠 = 𝑢𝑢𝑢𝑢 + (1/2) 𝑎𝑎𝑡𝑡 2 . (Fig.5).

(iii) Rectangular Hyperbola

This graph is represented by the equation 𝑥𝑥𝑥𝑥 = 𝐶𝐶 2 . If we plot graph between two quantities in inverse proportion, we get a rectangular hyperbola e.g. pressure and volume of an ideal gas at constant temperature, equation : 𝑃𝑃𝑃𝑃 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶. (Fig. 6) (iv) Exponential

When two quantities are related by an exponential equation, we get this graph, equation : 𝑦𝑦 = 𝑒𝑒 −𝑎𝑎𝑎𝑎 e.g. graph between the number of atoms remaining to undergo radioactive disintegration and time (Fig. 7). (v) Other graphs

(a) When a quantity decreases and then increases : The graph of angle of incidence and angle of deviation produced by a prism is an example of this case (Fig. 8). (b) When a quantity increases and remains constant : The graph of velocity and time, of a body starting with uniform acceleration and attaining uniform velocity (Fig. 9) is a very good example of this case.

(c) When a quantity decreases and remains constant : The graph of speed and time, of a body starting with an initial speed and uniform retardation, reaching a constant final speed, is a good example of this case (Fig. 10).

Displacement – time graphs :

a) The slope of displacement – time graph gives the velocity. If the graph is a straight line parallel to time axis then the body is at rest. When the time is noted it has some initial displacement.

b) The two graphs “b” and “c” represent a body moving with constant velocity having some initial displacement and zero initial displacement respectively.

c) The graph is a parabola. It is for a body starting from rest and moving with uniform acceleration (i.e., increasing slope)

d) The graph is a parabola. It shown that the body is moves with retardation ( is decreasing slope)

e) The graph is a parabola. It shows that the body is moving with uniform acceleration. (since slope increases).

f)

The body is moving is opposite direction to the initial direction [since with increasing time displacement decreases.]

g) The graph is a parabola. It shows that the body moves with retardation up to 𝑡𝑡 = 𝑡𝑡1 then it moves with acceleration up to 𝑡𝑡 = 𝑡𝑡2 and reaches the initial point. Here 𝑡𝑡1 = 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑜𝑜𝑜𝑜 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡2 = 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑜𝑜𝑜𝑜 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 This shows a body projected vertically upwards.

Velocity – time graphs :

The slope of velocity time graph acceleration and the area under the velocity time graph gives the displacement. a) This graph is a straight line parallel to time axis represent a body moving with uniform velocity.

b) This graph is a straight line passing through origin and represents a body moving with uniform acceleration (i.e., slope is constant and positive) starting from rest.

c) This graph is a straight line having some intercept on velocity axis and represents a body moving with uniform acceleration starts with an initial velocity.

d) This graph is a parabola represents a body moving with increasing acceleration (i.e. slope increases)

e) This graph represents a body moving with uniform retardation starting with an initial velocity. The body comes to rest when 𝑡𝑡 = 𝑡𝑡1

f)

The velocity – time graph of a ball freely falling from a height and undergoing elastic collisions with the floor is as shown in fig f (consider velocity is positive during upward motion and is negative during downward motion) OA, QC, SE : Descent. BQ, DS : Ascent OP, QR : time of descent, PQ, RS : Time of ascent. AB, CD during collision with the floor only direction of velocity changes without change in magnitude.

g) This graph represents the inelastic collision with the floor. After successive collisions, the ball comes to rest.

h) The graph represents a body projected vertically upwards from the ground Here OA = DC = Magnitude of velocity

OB = time of ascent BD = time of descent Here distance traveled by the body = Area of∆ le OAB + A rea of ∆ le BCD = 2 Area of ∆ OAB Displacement traveled by the body = Area of ∆ OAB – Area of ∆ BCD = 0 i.

The graph represents a body projected vertically upwards from the top of a tower [From fig i] OA = Velocity with which it is projected upwards [+𝑣𝑣] 𝐷𝐷𝐷𝐷 = Velocity with which it crosses the top of a tower downwards. [−𝑣𝑣] 𝑄𝑄𝑄𝑄 = Velocity of body when it strikes the ground. [ − 𝑉𝑉1 ] The total distance traveled by the body = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑜𝑜𝑜𝑜 ∆ 𝑂𝑂𝑂𝑂𝑂𝑂 + area of ∆ 𝐵𝐵𝐵𝐵𝐵𝐵 Displacement of the body = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑜𝑜𝑜𝑜 ∆ 𝑂𝑂𝑂𝑂𝑂𝑂 – area of ∆ 𝐵𝐵𝐵𝐵𝐵𝐵 = − 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑜𝑜𝑜𝑜 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 Height of tower = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑜𝑜𝑜𝑜 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

Acceleration – time graph : a.

This graph represents a body moving with uniform acceleration.

b.

The graph represents a body moving with non-uniform acceleration. The area of (𝑎𝑎 − 𝑡𝑡) graph with time axis is equal to change in velocity.