Question

A batter hits a baseball m = 0.41 kg from rest with a force of f = 75 n. The ball is in contact with the bat for t = 0.16 s. It is observed that the ball leaves the bat with an angle of θ = 24 degrees with respect to the horizontal. What is the initial velocity of the ball?

Answer 1

To find the initial velocity of the ball, we calculate the impulse using the average force and contact time, and then divide by the mass of the ball to get the initial velocity magnitude. The velocity components are then found using trigonometry, considering the angle with respect to the horizontal.

Step-by-step explanation:

To calculate the initial velocity of the baseball after it has been hit by the bat, we can use the impulse-momentum theorem, which states that the change in momentum, or impulse, is the product of the average force and the contact time. The formula is:

J = F × t

where J is the impulse, F is the average force, and t is the time the force is applied.

First, calculate the impulse:

J = 75 N × 0.16 s = 12 kg·m/s

Since the ball starts from rest, the change in momentum is equal to its final momentum. Therefore, we can use the formula:

p = m × v

Impulse is also the change in momentum (Δp), so:

12 kg·m/s = 0.41 kg × v

Solving for v, we get:

v = 12 kg·m/s / 0.41 kg ≈ 29.27 m/s

This is the magnitude of the velocity. To find the velocity components:

vx = v × cos(θ) = 29.27 m/s × cos(24°)

vy = v × sin(θ) = 29.27 m/s × sin(24°)

The initial velocity of the ball is the vector sum of these components.