A baseball pitcher throws a baseball with a speed of 43 m/s. Estimate the average acceleration of the ball during the throwing motion. In throwing the baseball, the pitcher accelerates it through a displacement of about 3.5 m, from behind the body to the point where it is released (Fig. 2–44).
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Verified step by step guidance

1. Identify the known quantities: the final velocity of the baseball \( v_f = 43 \; \text{m/s} \), the initial velocity \( v_i = 0 \; \text{m/s} \) (assuming the ball starts from rest), and the displacement \( d = 3.5 \; \text{m} \). The goal is to find the average acceleration \( a \).
2. Use the kinematic equation that relates acceleration, initial velocity, final velocity, and displacement: \( v_f^2 = v_i^2 + 2ad \). Rearrange this equation to solve for acceleration \( a \): \( a = \frac{v_f^2 - v_i^2}{2d} \).
3. Substitute the known values into the equation: \( a = \frac{(43)^2 - (0)^2}{2(3.5)} \).
4. Simplify the numerator \( (43)^2 \) and the denominator \( 2(3.5) \) to calculate the average acceleration.
5. The result will give the average acceleration of the baseball during the throwing motion in units of \( \text{m/s}^2 \).

Key Concepts

Acceleration

Acceleration is defined as the rate of change of velocity of an object with respect to time. It can be calculated using the formula a = (v_f - v_i) / t, where v_f is the final velocity, v_i is the initial velocity, and t is the time taken for the change. In this scenario, the pitcher accelerates the baseball from rest (0 m/s) to a speed of 43 m/s over a distance of 3.5 m.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this problem, kinematic equations can be used to relate the distance the baseball travels, its initial and final velocities, and the acceleration during the throw.

Displacement

Displacement is a vector quantity that refers to the change in position of an object. It is defined as the shortest distance from the initial to the final position, along with the direction. In the context of the baseball throw, the displacement of 3.5 m represents the distance over which the pitcher accelerates the ball before release, which is crucial for calculating the average acceleration.