To find the speed at which the Ping-Pong ball must move to have the same momentum as the bowling ball, we can start by calculating the momentum of the bowling ball.
The momentum \( p \) is given by the formula:
\[ p = m \cdot v \]
where \( m \) is the mass and \( v \) is the velocity.
For the bowling ball:
- Mass \( m_{\text{bowling}} = 10.8 , \text{kg} \)
- Velocity \( v_{\text{bowling}} = 3.12 , \text{m/s} \)
Calculating the momentum of the bowling ball:
\[ p_{\text{bowling}} = m_{\text{bowling}} \cdot v_{\text{bowling}} = 10.8 , \text{kg} \cdot 3.12 , \text{m/s} = 33.696 , \text{kg m/s} \]
Now, we want the momentum of the Ping-Pong ball to be equal to that of the bowling ball:
\[ p_{\text{Ping-Pong}} = p_{\text{bowling}} = 33.696 , \text{kg m/s} \]
Let \( m_{\text{Ping-Pong}} = 3.789 , \text{g} = 3.789 \times 10^{-3} , \text{kg} \) (since we need to convert grams to kilograms).
Let \( v_{\text{Ping-Pong}} \) be the unknown velocity we need to find. Setting the momentum of the Ping-Pong ball equal to the momentum of the bowling ball:
\[ m_{\text{Ping-Pong}} \cdot v_{\text{Ping-Pong}} = 33.696 , \text{kg m/s} \]
Now, substitute the mass of the Ping-Pong ball:
\[ (3.789 \times 10^{-3} , \text{kg}) \cdot v_{\text{Ping-Pong}} = 33.696 , \text{kg m/s} \]
Now, solve for \( v_{\text{Ping-Pong}} \):
\[ v_{\text{Ping-Pong}} = \frac{33.696 , \text{kg m/s}}{3.789 \times 10^{-3} , \text{kg}} \]
Calculating this gives:
\[ v_{\text{Ping-Pong}} \approx \frac{33.696}{0.003789} \approx 8895.10 , \text{m/s} \]
Therefore, the speed at which the Ping-Pong ball must move is approximately:
\[ \boxed{8895.10 , \text{m/s}} \]
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