A 0.311 kg tennis racket moves east at $$ 30.3 \mathrm{~m} / \mathrm{s} $$. A 0.0570 kg ball moves east at $$ 19.2 \mathrm{~m} / \mathrm{s} $$. If the racket and the ball have an elastic collision, what would the velocity of the racket be afterwards? $$ v_{1 f}=[?] \mathrm{m} / \mathrm{s} $$ Remember East is positive ( + ) : west is negative ( - ).

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To determine the final velocity of the tennis racket ($$ v_{1f} $$) after an elastic collision with the ball, we'll use the principles of **conservation of momentum** and **conservation of kinetic energy** applicable to elastic collisions.

### Given:
- **Mass of the racket ($$ m_1 $$)**: 0.311 kg
- **Initial velocity of the racket ($$ v_{1i} $$)**: +30.3 m/s (east)
- **Mass of the ball ($$ m_2 $$)**: 0.0570 kg
- **Initial velocity of the ball ($$ v_{2i} $$)**: +19.2 m/s (east)

### Formula for Final Velocities in Elastic Collisions:
For two objects undergoing an elastic collision in one dimension, the final velocities ($$ v_{1f} $$ and $$ v_{2f} $$) are given by:

$$
v_{1f} = \frac{(m_1 - m_2)}{(m_1 + m_2)} v_{1i} + \frac{2m_2}{(m_1 + m_2)} v_{2i}
$$

$$
v_{2f} = \frac{2m_1}{(m_1 + m_2)} v_{1i} + \frac{(m_2 - m_1)}{(m_1 + m_2)} v_{2i}
$$

### Calculating $$ v_{1f} $$:

Plugging in the known values:

$$
v_{1f} = \left( \frac{0.311 \, \text{kg} - 0.0570 \, \text{kg}}{0.311 \, \text{kg} + 0.0570 \, \text{kg}} \right) \times 30.3 \, \text{m/s} + \left( \frac{2 \times 0.0570 \, \text{kg}}{0.311 \, \text{kg} + 0.0570 \, \text{kg}} \right) \times 19.2 \, \text{m/s}
$$

$$
v_{1f} = \left( \frac{0.254 \, \text{kg}}{0.368 \, \text{kg}} \right) \times 30.3 \, \text{m/s} + \left( \frac{0.114 \, \text{kg}}{0.368 \, \text{kg}} \right) \times 19.2 \, \text{m/s}
$$

$$
v_{1f} \approx 0.69 \times 30.3 \, \text{m/s} + 0.31 \times 19.2 \, \text{m/s}
$$

$$
v_{1f} \approx 20.9 \, \text{m/s} + 5.95 \, \text{m/s} = 26.85 \, \text{m/s}
$$

### Final Answer:
$$
v_{1f} \approx 26.9 \, \mathrm{m/s} \, \text{(east)}
$$
After the collision, the racket moves east at approximately 26.9 m/s.

A 0.311 kg tennis racket moves east at .
A 0.0570 kg ball moves east at .
If the racket and the ball have an elastic
collision, what would the velocity of
the racket be afterwards?

Remember East is positive ( + ) : west is negative ( - ).

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A 0.311 kg tennis racket moves east at . A 0.0570 kg ball moves east at . If the racket and the ball have an elastic collision, what would the velocity of the racket be afterwards? Remember East is positive ( + ) : west is negative ( - ).