Sup 4

In putting, the force with which a golfer strikes a ball is planned so that the ball will stop within some small distance of the cup, say 1.0 m long or short, in case the putt is missed. Accomplishing this from an uphill lie (that is, putting downhill) is more difficult than from a downhill lie. To see why, assume that on a particular green the ball decelerates constantly at 2.0 m/s² going downhill, and constantly at 3.0 m/s² going uphill. Suppose we have an uphill lie 7.0 m from the cup. Calculate the allowable range of initial velocities we may impart to the ball so that it stops in the range 1.0m short to 1.0m long of the cup. Do the same for a downhill lie 7.0 m from the cup. What in your results suggests that the downhill putt is more difficult? Let’s consider the uphill putt (the “downhill lie”) first. We will apply the gang-of-five equation v 2f = v02 + 2aΔx . We are interested in the position of the ball as it comes to rest; that is, as v f = 0 .

We furthermore know that a = −3.0 sm2 . We want the change in position to be between 6.0 m and 8.0 m; to this end, we will solve the equation for v0 twice. First, we will solve 0 = v02 + (2)(−3.0)(6.0) and then we will solve 0 = v02 + (2)(−3.0)(8.0) . It ends up that in the first case (falling one meter short), v0 = 6 ms , and in the second case (overshooting by one meter), v0 = 6.93 ms . We can, of course, have any velocity between these two bounds, because an intermediate velocity will place the ball somewhere between one meter short and one meter long. Our range of allowable velocities is, therefore, 0.93 ms . Let us now consider the downhill putt (the “uphill lie”). We once again apply the same equation, but with the caveat that a = −2.0 sm2 . We once again solve our equation twice: 0 = v02 + (2)(−2.0)(6.0) and 0 = v02 + (2)(−2.0)(8.0) . This gives us bounds of v0 = 4.90 ms and v0 = 5.66 ms . This gives us a range of allowable velocities of 0.76 ms . The fact that the range of allowable velocities for the downhill putt is smaller than the range for the uphill putt seems to indicate that it takes more finesse to ensure that the bell is launched with a velocity in that range, causing the putt to be more difficult. Quod erat demonstratum.