A 0.145-kg baseball pitched horizontally at 35.0 m/s strikes a bat and pops straight up to a height of 31.5 m. If the contact time between bat and ball is 2.5 ms, calculate the average force between the ball and bat during contact.

Verified step by step guidance

1. Step 1: Identify the initial and final velocities of the baseball. The initial velocity is given as 35.0 m/s horizontally. After the collision, the ball pops straight up, so its final vertical velocity can be determined using the kinematic equation for motion under gravity: $\sqrt{2gh}$, where $g$ is the acceleration due to gravity (9.8 m/s²) and $h$ is the height (31.5 m).
2. Step 2: Calculate the change in velocity. The initial velocity is horizontal, and the final velocity is vertical. Use vector addition to find the magnitude of the change in velocity: $\sqrt{v^2+u^2}$, where $v$ is the final vertical velocity and $u$ is the initial horizontal velocity.
3. Step 3: Determine the change in momentum. Momentum is given by $p=mv$, where $m$ is the mass of the baseball (0.145 kg) and $v$ is the velocity. Calculate the change in momentum using the magnitude of the change in velocity from Step 2.
4. Step 4: Use the impulse-momentum theorem to relate the change in momentum to the average force. The theorem states $F\Delta t=\Delta p$, where $F$ is the average force, $\Delta t$ is the contact time (2.5 ms or 0.0025 s), and $\Delta p$ is the change in momentum. Rearrange to solve for $F$: $F=\frac{\Delta }{p}$.
5. Step 5: Substitute the values for the change in momentum and contact time into the equation from Step 4 to calculate the average force. Ensure all units are consistent (e.g., kilograms, meters per second, seconds).

Key Concepts

Momentum

Momentum is the product of an object's mass and its velocity, represented by the equation p = mv. In this scenario, the baseball's momentum before and after the collision with the bat is crucial for understanding the changes in motion. The change in momentum during the impact can help determine the average force exerted by the bat on the ball.

Impulse

Impulse is defined as the change in momentum of an object when a force is applied over a period of time, expressed as J = FΔt. In this case, the impulse experienced by the baseball during its brief contact with the bat will allow us to calculate the average force exerted. The relationship between impulse and force is essential for solving the problem.

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. In this problem, kinematic equations can be used to analyze the vertical motion of the baseball after it leaves the bat, particularly to find the initial velocity needed to reach a height of 31.5 m. Understanding kinematics is vital for connecting the motion of the baseball to the forces involved.