Questions: A 57.0 g tennis ball is thrown with a speed of 35.0 m/s at a 7.3 kg bowling ball that is moving at a speed of 1.0 m/s towards the tennis ball. The two collide and the tennis ball bounces back in the opposite direction as it previously moving at a speed of 20.0 m/s. Determine the speed of the bowling ball after the collision. Was the collision elastic, inelastic, or completely inelastic?

Solution Steps

We have a collision between a tennis ball and a bowling ball. The masses and initial velocities are given:

• Mass of tennis ball, \( m_1 = 57.0 \, \text{g} = 0.057 \, \text{kg} \)
• Initial velocity of tennis ball, \( v_{1i} = 35.0 \, \text{m/s} \)
• Mass of bowling ball, \( m_2 = 7.3 \, \text{kg} \)
• Initial velocity of bowling ball, \( v_{2i} = -1.0 \, \text{m/s} \) (negative because it is moving towards the tennis ball)
• Final velocity of tennis ball, \( v_{1f} = -20.0 \, \text{m/s} \) (negative because it bounces back)

We need to find the final velocity of the bowling ball, \( v_{2f} \), and determine the type of collision.

The law of conservation of momentum states that the total momentum before the collision equals the total momentum after the collision.

\[ m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \]

Substitute the known values:

\[ (0.057 \, \text{kg})(35.0 \, \text{m/s}) + (7.3 \, \text{kg})(-1.0 \, \text{m/s}) = (0.057 \, \text{kg})(-20.0 \, \text{m/s}) + (7.3 \, \text{kg})v_{2f} \]

Calculate the left side:

\[ 1.995 - 7.3 = -1.14 + 7.3v_{2f} \]

Simplify and solve for \( v_{2f} \):

\[ -5.305 = -1.14 + 7.3v_{2f} \]

\[ 7.3v_{2f} = -5.305 + 1.14 \]

\[ 7.3v_{2f} = -4.165 \]

\[ v_{2f} = \frac{-4.165}{7.3} \approx -0.5705 \, \text{m/s} \]

To determine if the collision is elastic, we need to check if kinetic energy is conserved.

Calculate initial kinetic energy:

\[ KE_{\text{initial}} = \frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 \]

\[ = \frac{1}{2} (0.057)(35.0)^2 + \frac{1}{2} (7.3)(1.0)^2 \]

\[ = 34.9125 + 3.65 = 38.5625 \, \text{J} \]

Calculate final kinetic energy:

\[ KE_{\text{final}} = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2 \]

\[ = \frac{1}{2} (0.057)(20.0)^2 + \frac{1}{2} (7.3)(-0.5705)^2 \]

\[ = 11.4 + 1.1875 \approx 12.5875 \, \text{J} \]

Since \( KE_{\text{initial}} \neq KE_{\text{final}} \), the collision is inelastic.

Final Answer