What is the speed of the ball after a collision with a baseball bat, given the bat's mass is mbat, the ball's mass is mball, the bat's initial velocity is 36 m/s, and the ball's initial velocity is -20 m/s?

Answer 1

To determine the speed of the ball after the collision with the bat, we can use the principle of conservation of momentum. For this scenario, we will assume a perfectly elastic collision where both momentum and kinetic energy are conserved.

Given:

• Mass of bat, \( m_{\text{bat}} \)
• Mass of ball, \( m_{\text{ball}} \)
• Initial velocity of the bat, \( v_{\text{bat}_i} = 36 , \text{m/s} \) (bat moving forward)
• Initial velocity of the ball, \( v_{\text{ball}_i} = -20 , \text{m/s} \) (ball moving backward)

Conservation of Momentum Equation:

The momentum before the collision equals the momentum after the collision.

\[ m_{\text{bat}} \cdot v_{\text{bat}i} + m{\text{ball}} \cdot v_{\text{ball}i} = m{\text{bat}} \cdot v_{\text{bat}f} + m{\text{ball}} \cdot v_{\text{ball}_f} \]

Where:

• \( v_{\text{bat}_f} \): final velocity of the bat
• \( v_{\text{ball}_f} \): final velocity of the ball

Assumptions:

We will assume the bat doesn't change its speed significantly upon impact and we might consider it approximately constant for small mass ratios, or alternatively, we could derive the ball's final speed based on typical elastic collision physics.

Elastic Collision:

In a perfectly elastic collision, we can also use the following formula for the final velocities of two objects (the baseball bat and ball):

\[ v_{\text{ball}f} = \frac{(m{\text{ball}} - m_{\text{bat}})}{(m_{\text{ball}} + m_{\text{bat}})} v_{\text{ball}i + v{\text{bat}i}} + \frac{2 m{\text{bat}}}{(m_{\text{ball}} + m_{\text{bat}})} v_{\text{bat}_i} \]

Solving the final velocities would explicitly require the masses of the bat and the ball.

However, based on a simpler and typically used approximation for ball-bat collisions, where \(m_{\text{bat}}\) is considerably larger than \(m_{\text{ball}}\):

Approximate Final Velocity of Ball:

Assuming the masses are such that \(m_{\text{bat}} >> m_{\text{ball}}\): \[ v_{\text{ball}f} \approx \frac{-v{\text{ball}i} + v{\text{bat}_i}}{2} \]

Thus, \[ v_{\text{ball}_f} = \frac{20 + 36}{2} = \frac{56}{2} = 28 , \text{m/s} \]

Conclusion

Therefore, if the mass of the bat is significantly larger than that of the ball, the velocity of the ball after it collides with the bat is approximately \( 28 , \text{m/s} \) in the positive direction (the same direction as the bat). For specific applications, the mass values will provide the most accurate result.