Question

A baseball player uses a bat with mass \(m_{\text {bat }}\) to hit a ball with mass \(m_{\text {ball }}\). Right before he hits the ball, the bat's initial velocity is \(35.0 \mathrm{~m} / \mathrm{s}\), and the ball's initial velocity is \(-30.0 \mathrm{~m} / \mathrm{s}\) (the positive direction is along the positive \(x\) -axis). The bat and ball undergo a one-dimensional elastic collision. Find the speed of the ball after the collision. Assume that \(m_{\text {bat }}\) is much greater than \(m_{\text {ball }}\), so the center of mass of the two objects is essentially at the bat.

Short answer

Answer: The final velocity of the ball after the collision is approximately 5.0 m/s in the positive x-axis direction.

Step-by-step solution

Write down the conservation of momentum formula

In a one-dimensional elastic collision, the total momentum before the collision is equal to the total momentum after the collision. This is expressed mathematically as: \(m_{\text {bat}}v_{\text {bat_initial}} + m_{\text {ball}}v_{\text {ball_initial}} = m_{\text {bat}}v_{\text {bat_final}} + m_{\text {ball}}v_{\text {ball_final}}\) Where \(v\) represents the velocities of the bat and ball, and the subscripts "initial" and "final" represent before and after the collision, respectively.

Write down the conservation of kinetic energy formula

In a one-dimensional elastic collision, the total kinetic energy before the collision is also equal to the total kinetic energy after the collision. This is expressed mathematically as: \(0.5m_{\text {bat}}v_{\text {bat_initial}}^2 + 0.5m_{\text {ball}}v_{\text {ball_initial}}^2 = 0.5m_{\text {bat}}v_{\text {bat_final}}^2 + 0.5m_{\text {ball}}v_{\text {ball_final}}^2\)

Use the given conditions

We are given that \(m_{\text {bat}}\) is much greater than \(m_{\text {ball}}\), so the center of mass of the two objects is essentially at the bat. This means that the final velocity of the bat essentially does not change after the collision. Therefore, we can set \(v_{\text {bat_final}}\) equal to \(v_{\text {bat_initial}}\), which is \(35.0 \mathrm{~m} / \mathrm{s}\).

Use the conservation of momentum formula

We can now apply the conservation of momentum formula with the given initial velocities and the assumption that the final velocity of the bat does not change: \(m_{\text {bat}}(35.0) + m_{\text {ball}}(-30.0) = m_{\text {bat}}(35.0) + m_{\text {ball}}v_{\text {ball_final}}\) Now, rearrange the equation to solve for \(v_{\text {ball_final}}\): \(v_{\text {ball_final}} = \frac{m_{\text {bat}}(35.0) - m_{\text {ball}}(-30.0)}{m_{\text {ball}}}\) Since \(m_{\text {bat}}\) is much greater than \(m_{\text {ball}}\), \(m_{\text {bat}}(35.0)\) is a much larger term, which makes the ratio of the two masses in the equation much greater than one. This means that the final velocity of the ball will be essentially the sum of the initial velocities: \(v_{\text {ball_final}} \approx 35.0 + (-30.0) = 5.0 \mathrm{~m} / \mathrm{s}\) So, the speed of the ball after the collision will be approximately 5.0 m/s in the positive \(x\)-axis direction.

Key concepts

Elastic Collision

An elastic collision is a type of collision where both momentum and kinetic energy are conserved. In simpler terms, after the collision, the total energy and momentum stay the same as before.

• Elastic collisions occur when two objects collide and then bounce away from each other without any lasting deformation or heat generation.


• This means the objects retain almost all their initial kinetic energy after the collision.

Understanding elastic collisions is beneficial to analyze how different objects interact, especially when it comes to changes in their speeds and directions after they have collided. It's like playing billiards, where the balls hit each other and move away swiftly, demonstrating almost perfect elastic collisions.
During an elastic collision, the relative speed between the two objects remains constant in magnitude, but the direction can change. Thus, observing an elastic collision will show the two objects exchanging their velocities, which is exactly what happens in our baseball player scenario.

Kinetic Energy

Kinetic energy is the energy an object has due to its motion. Any object that is moving, whether it's a tiny ball in a baseball game or a comet in space, has kinetic energy.

• It is calculated using the formula: \( KE = 0.5 \cdot m \cdot v^2 \).


• Here, \( m \) is the mass of the object, and \( v \) is its velocity.

In the context of our problem, the kinetic energies of both the bat and the ball are central to determining what happens after the collision. An elastic collision, like the one described, ensures that the sum of the kinetic energies before the collision equals the sum after.
This balance ensures no energy is lost, letting us predict the outcome quite confidently using formulas. So, when the player swings the bat, the energy from the swing is transferred to the ball, affecting its velocity dramatically. The excitement in this problem is realizing that energy conservation allows us to calculate outcomes in such dynamic and fast interactions!

Center of Mass

The center of mass is a point that represents the average position of all the mass in an object or system of objects. It's effectively the balance point, like the middle of a seesaw.

• In our problem, the bat is the heavier object, meaning its center of mass dominates the scenario.


• For this exercise, the center of mass could be assumed to lie within the bat due to its greater mass.

When solving collision problems like these, knowing about the center of mass helps us understand why certain simplifications are possible. For instance, since the bat is so much heavier compared to the ball, its velocity doesn’t change significantly, letting us approximate complex dynamics more easily.
If you've ever spun a ping pong ball on a string with a heavy rock at the other end, you'd have noticed the ball whips around wildly while the rock stays pretty still. That's the center of mass concept in action, dictating movement and stability.