To find the final speed of the red ball after the collision, we can use the principle of conservation of momentum. The total momentum before the collision must equal the total momentum after the collision.
Let:
- \( m_g = 0.50 , \text{kg} \) (mass of the green ball)
- \( m_r = 0.50 , \text{kg} \) (mass of the red ball)
- \( v_g = 10.5 , \text{m/s} \) (initial speed of the green ball)
- \( v_r = 0 , \text{m/s} \) (initial speed of the red ball before the collision)
- \( v_g' = 0 , \text{m/s} \) (final speed of the green ball after the collision)
- \( v_r' \) (final speed of the red ball after the collision)
The conservation of momentum principle can be expressed mathematically as follows:
\[ m_g v_g + m_r v_r = m_g v_g' + m_r v_r' \]
Substituting in the known values:
\[ (0.50 , \text{kg})(10.5 , \text{m/s}) + (0.50 , \text{kg})(0 , \text{m/s}) = (0.50 , \text{kg})(0 , \text{m/s}) + (0.50 , \text{kg})(v_r') \]
This simplifies to:
\[ 5.25 , \text{kg m/s} = 0 + (0.50 , \text{kg}) (v_r') \]
Now we can isolate \( v_r' \):
\[ 5.25 , \text{kg m/s} = 0.50 , \text{kg} \cdot v_r' \]
\[ v_r' = \frac{5.25 , \text{kg m/s}}{0.50 , \text{kg}} = 10.5 , \text{m/s} \]
So, the final speed of the red ball after the collision is:
\[ v_r' = 10.5 , \text{m/s} \]
Thus, the answer is (d) 10.5 m/s.
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