Chapter 2, Problem 67GP

To determine

The allowable ranges of initial velocities a golfer may impart to the ball from the uphill and downhill.

To explain: The reason for the considering downhill putt more difficult.

Answer to Problem 67GP

For uphill lie, initial velocity for $1.0\text{m}$ short to the cup is $4.9\text{m/s}$ and for $1.0\text{m}$ long is $5.7\text{m/s}$ . For downhill lie, the initial velocity for $1.0\text{m}$ short to the cup is $6.0\text{m/s}$ and for $1.0\text{m}$ long is $6.9\text{m/s}$ In putting the ball, more precision and control is required. Thus, downhill putting is difficult comparatively.

Explanation of Solution

Given:

The given situation is shown below.

  

The final velocity of the ball in both the cases is $v=0\text{m/s}$ . Deceleration of the ball in moving downwards from uphill is $a=-2{\text{m/s}}^2$ and in moving upwards from downhill is $a=-3{\text{m/s}}^2$ . Distance when the ball needs to be stopped $1\text{m}$ short to the cup is $s_1=6\text{m}$ . Distance when the ball needs to be stopped $1\text{m}$ long of the cup is $s_2=8\text{m}$ .

Formula used:

Using Newton’s third law of motion,

  $v^2=u^2+2as$

Calculation:

Calculate for the uphill lie.

The initial velocity of the ball when it is $1\text{m}$ short to the cup is

  $\begin{array}{c}u=\sqrt{v^2-2a{s}_1}\\ =\sqrt{{\left(0\right)}^2-2\times \left(-2\right)\times 6}=4.9\text{m/s}\end{array}$

The initial velocity of the ball when it is $1\text{m}$ long to the cup is

  $\begin{array}{c}u=\sqrt{v^2-2a{s}_2}\\ =\sqrt{{\left(0\right)}^2-2\times \left(-2\right)\times 8}=5.7\text{m/s}\end{array}$

Calculate for the downhill lie.

The initial velocity of the ball when it is $1\text{m}$ short to the cup is

  $\begin{array}{c}u=\sqrt{v^2-2a{s}_1}\\ =\sqrt{{\left(0\right)}^2-2\times \left(-3\right)\times 6}=6.0\text{m/s}\end{array}$

The initial velocity of the ball when it is $1\text{m}$ long to the cup is

  $\begin{array}{c}u=\sqrt{v^2-2a{s}_2}\\ =\sqrt{{\left(0\right)}^2-2\times \left(-3\right)\times 8}=6.9\text{m/s}\end{array}$

The spread of acceptable velocities for the uphill is $5.7-4.9=0.8\text{m/s}$

The spread of acceptable velocities for the downhill is $6.9-6.0=0.9\text{m/s}$

Conclusion:

Thus, the initial velocity of the ball when ball is being putted down the hill for $1\text{m}$ short to the cup is $4.9\text{m/s}$ and for $1\text{m}$ long to the cup is $5.7\text{m/s}$ . Thus, initial velocity of the ball when ball is being putted up the hill for $1\text{m}$ short to the cup is $6.0\text{m/s}$ and for $1\text{m}$ long to the cup is $6.9\text{m/s}$ . The range of initial velocities of the ball being putted down the hill is lesser compared to up the hill. As more precision and control is necessary in putting the ball. Hence, downhill putt is more difficult.