In putting, the force with which a golfer strikes a ball is planned so that at the ball will stop within some small distance of the cup, say 1.0m long or short, in case the putt is missed. Accomplishing this from an uphill lie(that is, putting the ball downhill, see Fig. 2-48) is more difficult that from a downhill lie. To see why, assume that on a particular green the ball decelerates constantly at 1.8m/s2 going downhill, and constantly at 2.8m/s2 going uphill. Suppose we have an uphill lie 7.0m from the cup. Calculate the allowable range of initial velocities we may impart to the ball so that is stops in the range 1.0 short to 1.0m long of the cup. Do the same for a downhill lie 7.0m from the cup. What is your results suggests that the downhill putt is more difficult?
To solve this problem, we first need to understand the physics involved in the motion of the golf ball. We know that the ball decelerates constantly during its motion, and we are given the deceleration values for both uphill and downhill putts.
Let's consider the uphill putt first. We are given that the ball decelerates at a rate of 2.8 m/s^2 going uphill. We need to find the allowable range of initial velocities that will allow the ball to stop within the range of 1.0m short to 1.0m long of the cup, which is 7.0m away.
To solve this, we can use the equations of motion. The relevant equation for this scenario is:
vf^2 = vi^2 + 2ad
Where:
vf = final velocity (0 m/s since the ball needs to stop)
vi = initial velocity (what we need to solve for)
a = acceleration (we are given 2.8 m/s^2 for uphill putts)
d = distance traveled (7.0 m for both uphill and downhill putts)
Rearranging the equation, we get:
vi^2 = -2ad
Substituting the given values:
vi^2 = -2 * 2.8 * 7.0
Simplifying:
vi^2 = -39.2
Taking the square root:
vi ≈ ±6.26 m/s
Therefore, the allowable range of initial velocities for an uphill lie is approximately -6.26 m/s to +6.26 m/s.
Now let's consider the downhill putt. We are given that the ball decelerates at a rate of 1.8 m/s^2 going downhill. Using the same equation and substituting the values:
vi^2 = -2 * 1.8 * 7.0
Simplifying:
vi^2 = -25.2
Taking the square root:
vi ≈ ±5.02 m/s
Therefore, the allowable range of initial velocities for a downhill lie is approximately -5.02 m/s to +5.02 m/s.
Comparing the two results, we can see that the allowable range of initial velocities for the uphill putt is larger (in both negative and positive values) than that of the downhill putt. This suggests that it is more difficult to control the speed and distance of the ball while putting from an uphill lie.