Question

A ballplayer standing at home plate hits a baseball that is caught by another player at the same height above the ground from which it was hit. The ball is hit with an initial velocity of \(22.0 \mathrm{m} / \mathrm{s}\) at an angle of \(60.0^{\circ}\) above the horizontal. (a) How high will the ball rise? (b) How much time will elapse from the time the ball leaves the bat until it reaches the fielder? (c) At what distance from home plate will the fielder be when he catches the ball?

Short answer

(b) What is the time elapsed from the time the ball leaves the bat until it reaches the fielder? (c) What is the distance from home plate to the fielder when they catch the ball? Answer: Follow the step-by-step solution provided to calculate the values for (a) maximum height, (b) time elapsed, and (c) distance from home plate to the fielder when they catch the ball.

Step-by-step solution

Calculate the initial horizontal and vertical velocities

We have the initial velocity, \(v_0 = 22.0 \mathrm{m} / \mathrm{s}\), and launch angle, \(\theta = 60.0^{\circ}\). We will find the initial horizontal velocity, \(v_{0x}\), and vertical velocity, \(v_{0y}\), using the trigonometric functions: \(v_{0x} = v_0 \cos\theta\) \(v_{0y} = v_0 \sin\theta\)

Calculate the time of flight

Since the ball is caught at the same height it was hit, we know that the vertical displacement will be zero. Applying the kinematic equation for vertical motion: \(0 = v_{0y}t - \frac{1}{2}gt^2\) Solve this equation for the time, \(t\). Where \(g\) is the acceleration due to gravity, \(g = 9.81 \mathrm{m} / \mathrm{s^2}\).

Calculate the maximum height

To find the maximum height, we can use the following kinematic equation: \(h = v_{0y}t_1 - \frac{1}{2}gt_1^2\) Where \(h\) is the maximum height and \(t_1\) is the time to reach the maximum height. Since the vertical velocity is zero at the maximum height, we can use: \(v_y = v_{0y} - gt_1\) Find \(t_1\) and then substitute it into the equation for \(h\).

Calculate the horizontal distance

Lastly, we can find the horizontal distance, also known as the range, using the initial horizontal velocity and the time of flight: \(R = v_{0x}t\) Where \(R\) is the horizontal distance or range. Now we can answer the questions: (a) The maximum height the ball will rise is the value of \(h\) calculated in Step 3. (b) The time elapsed from the time the ball leaves the bat until it reaches the fielder is the value of \(t\) calculated in Step 2. (c) The distance from home plate the fielder will be when they catch the ball is the value of \(R\) calculated in Step 4.