(II) 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. Determine the initial momentum of the baseball before contact with the bat. Use the formula for momentum: $p=mv$, where $m$ is the mass of the baseball (0.145 kg) and $v$ is its velocity (35.0 m/s).
2. Calculate the final velocity of the baseball after it pops straight up to a height of 31.5 m. Use the kinematic equation: $v^2=v^0+2gh$, where $v$ is the final velocity (0 m/s at the top of the motion), $g$ is the acceleration due to gravity (9.8 m/s²), and $h$ is the height (31.5 m). Solve for the velocity just after contact with the bat.
3. Determine the change in momentum of the baseball during the collision. The change in momentum is given by: $\Delta p=m(v{f}_y-v_x)$, where $v_x$ is the initial horizontal velocity and $v_y$ is the vertical velocity after contact.
4. Calculate the average force exerted on the baseball during the collision. Use the impulse-momentum theorem: $F{\mathrm{avg}}_{}=\Delta p/\Delta t$, where $\Delta p$ is the change in momentum and $\Delta t$ is the contact time (2.5 ms or 0.0025 s).
5. Substitute the values for $\Delta p$ and $\Delta t$ into the formula for average force to find the result.

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 change in motion. The change in momentum can be used to determine the impulse experienced by the ball, which is directly related to the average force exerted during the contact time.

Impulse

Impulse is defined as the change in momentum of an object when a force is applied over a period of time. It can be calculated using the formula Impulse = Force × Time. In this case, the average force exerted by the bat on the baseball can be determined by rearranging the impulse formula, considering the time of contact and the change in momentum of the baseball.

Kinetic Energy and Potential Energy

Kinetic energy is the energy of an object due to its motion, calculated as KE = 0.5mv², while potential energy is the energy stored due to an object's position, such as gravitational potential energy, given by PE = mgh. When the baseball pops up to a height of 31.5 m, its kinetic energy is converted into potential energy. Understanding this energy transformation is essential for analyzing the forces involved during the collision.