Chapter 6.3, Problem 44E

Solution

a.

To check: Whether the ball clear the fence.

No, the ball does not clear the fence.

Given information: The initial velocity of the ball is $120$ ft/sec and the ball hit when it is $4$ ft above the ground. The ball leaves the bat at an angle $30°$ and heads toward $30$ -ft fence $350$ ft from home plate.

Calculation:

Calculate the vertical velocity of the ball.

  $\begin{array}{c}{v}_y=120\sin 30°\\ =120\left(\frac{1}{2}\right)\\ =60\text{ft}/\mathrm{s}\end{array}$

Calculate the horizontal velocity of the ball.

  $\begin{array}{c}{v}_x=120\cos 30°\\ =120\left(\frac{\sqrt{3}}{2}\right)\\ =60\sqrt{3}\text{ft}/\mathrm{s}\end{array}$

Calculate the horizontal position at time $t$ .

  $s_x=v_{x}t$

Substitute $60\sqrt{3}$ for $v_x$ in the above equation.

  $s_x=60\sqrt{3}t$

Consider the equation for vertical position at time $t$ for initial position $v_0$ , height $h_0$ and acceleration for gravity $g$ .

  $s_y=-g{t}^2+v_{0}t+h_0$

The value of gravity is $16$ . Substitute $16$ for $6$ , $60$ for initial velocity, and $4$ for initial height.

  $s_y=-16t^2+60t+4$

Substitute $350$ for $v_x$ in the equation to calculate the time.

  $\begin{array}{c}350=60\sqrt{3}t\\ t=\frac{350}{60\sqrt{3}}\\ t\approx 3.4\text{s}\end{array}$

Substitute $3.4$ for $t$ in the equation $s_y=-16t^2+60t+4$ .

  $\begin{array}{c}{s}_y=-16{\left(3.4\right)}^2+60\left(3.4\right)+4\\ \approx 24.6\end{array}$

The height of the ball when it reaches the fence is $24.6$ ft. It is less than the height of the fence $30$ ft. So, the ball does not clear the fence.

Thus, the ball does not clear the fence.

b.

To find: The height by which ball clear the fence. If it does not clear the fence, could the ball be caught.

No, the ball cannot be caught.

Given information: The initial velocity of the ball is $120$ ft/sec and the ball hit when it is $4$ ft above the ground. The ball leaves the bat at an angle $30°$ and heads toward $30$ -ft fence $350$ ft from home plate.

Calculation:

From part (a), the height of the ball above the ground at the fence is $24.6$ ft.

Consider that the average height of a fielder is about $6$ ft and outstretch of an arm is about $3$ ft.

Calculate the height that a fielder should jump to catch the ball.

  $\begin{array}{c}24.6-\left(6+3\right)=24.6-9\\ =15.6\text{ft}\end{array}$

So, the ball cannot be caught.

Thus, the ball cannot be caught.