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 2.00m long or short, in case the putt is missed. Accomplishing this from an uphill lie (that is, putting the ball downhill, ) is more difficult than from a downhill lie.
part a)To see why, assume that on a particular green the ball decelerates constantly at 1.6m/s^2 going downhill, and constantly at 2.5m/s^2 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 it stops in the range 2.00m short to 2.00m long of the cup.
part b)Do the same for a downhill lie 7.0m from the cup.
Part a) To calculate the allowable range of initial velocities for an uphill lie, we need to consider the distance traveled by the ball before it comes to a stop.
Using the equations of motion, we know that the distance traveled by an object under constant deceleration is given by:
d = (v^2 - u^2) / (2a)
where d is the distance traveled, v is the final velocity (which is 0 in this case, as the ball comes to a stop), u is the initial velocity, and a is the acceleration.
In this case, the deceleration going uphill (a) is 2.5 m/s^2 and the distance to the cup (d) is 7.0 m. We want the ball to stop within a range of 2.00m long or short of the cup. Therefore, the range of distances within which the ball should stop is (7.0 + 2.0 + 2.0) = 11.0 m to (7.0 - 2.0 - 2.0) = 3.0 m.
Using the equation d = (v^2 - u^2) / (2a) and rearranging it to solve for u, we have:
u = sqrt(v^2 - 2ad)
Plugging in the values, we get:
u = sqrt(0^2 - 2 * 2.5 * 7.0) = sqrt(-35) (Note: The negative sign indicates that the ball cannot be struck uphill with this initial velocity.)
Therefore, there is no allowable range of initial velocities in this case because the square root of a negative number is not defined.
Part b) For a downhill lie, we can use the same equation, but with the downhill deceleration (a) of 1.6 m/s^2. The range of distances within which the ball should stop is (7.0 + 2.0 + 2.0) = 11.0 m to (7.0 - 2.0 - 2.0) = 3.0 m.
Using the equation u = sqrt(v^2 - 2ad), we have:
u = sqrt(0^2 - 2 * 1.6 * 7.0) = sqrt(-22.4) (Note: Again, the negative sign indicates that the ball cannot be struck downhill with this initial velocity.)
Therefore, similarly to the uphill lie, there is no allowable range of initial velocities for the downhill lie.
In both cases, due to the negative values under the square root, there is no valid solution for the range of initial velocities that would result in the ball stopping within the defined range from the cup.