Question

A baseball slugger hits a pitch and watches the ball fly into the bleachers for a home run, landing h=9.5m higher than it was hit. When visiting with the fan that caught the ball, he learned the ball was moving with final velocity vf=33.75m/s at an angle θf=32.5∘ below the horizontal when caught. Assume the ball encountered no air resistance, and use a Cartesian coordinate system with the origin located at the ball's initial position. Create an expression for the ball’s initial horizontal velocity, v0,x , in terms of the variables given in the problem statement. Calculate the magnitude, in meters per second, of the vertical component of the ball’s initial velocity.

Answer 1

To determine the initial horizontal velocity of the ball, we can equate it with the horizontal component of the final velocity due to the absence of air resistance.

Step-by-step explanation:

To find the ball’s initial horizontal velocity, v0,x, we need to use the information given about the ball's final position. Since there is no air resistance, the horizontal velocity remains constant throughout the flight of the ball. The horizontal component of the final velocity, vf,x, is given by vf × cos(θf), where θf is the angle below the horizontal.

Thus, v0,x = vf,x because horizontal velocity does not change when air resistance is ignored. To calculate the magnitude of the vertical component of the ball’s initial velocity, v0,y, we can use the vertical motion equations.

Rearrange the equation to solve for u, which represents the initial vertical velocity. Since the ball ends the motion 9.5 m higher than it was hit, and the final velocity is at an angle to the horizontal, we can calculate the final vertical velocity component, vf,y = vf × sin(θf), and then use the kinematic equation to solve for v0,y.