Answer :
To find the final velocity of the disk-stick system after the collision, we can use the principle of conservation of linear momentum. The total momentum before the collision (when only the disk is moving) should equal the total momentum after the collision (when the disk and stick move together).
1. Calculate the momentum before the collision:
The momentum \(p\) of the disk before the collision is:
[tex]\[ p_{\text{disk}} = m_{\text{disk}} \times v_{\text{initial}} \][/tex]
where
[tex]- \( m_{\text{disk}} = 51.7 \) g \(= 0.0517 \) kg (mass of the disk) - \( v_{\text{initial}} = 34.7 \) m/s (initial velocity of the disk) Therefore, \[ p_{\text{disk}} = 0.0517 \text{ kg} \times 34.7 \text{ m/s} = 1.7949 \text{ kg m/s} \][/tex]
2. Calculate the total mass of the system:
The total mass \( M \) of the disk-stick system is:
[tex]\[ M = m_{\text{disk}} + m_{\text{stick}} \] where - \( m_{\text{stick}} = 2.12 \) kg (mass of the stick) Therefore, \[ M = 0.0517 \text{ kg} + 2.12 \text{ kg} = 2.1717 \text{ kg} \][/tex]
3. Apply conservation of momentum:
According to the principle of conservation of linear momentum, the total momentum before the collision equals the total momentum after the collision.
[tex]\[ p_{\text{before}} = p_{\text{after}} \] \[ m_{\text{disk}} \times v_{\text{initial}} = (m_{\text{disk}} + m_{\text{stick}}) \times v_{\text{final}} \][/tex]
4. Solve for the final velocity [tex]\( v_{\text{final}} \)[/tex]:
Rearrange the equation to solve for[tex]\( v_{\text{final}} \): \[ v_{\text{final}} = \frac{m_{\text{disk}} \times v_{\text{initial}}}{m_{\text{disk}} + m_{\text{stick}}} \] \[ v_{\text{final}} = \frac{0.0517 \text{ kg} \times 34.7 \text{ m/s}}{0.0517 \text{ kg} + 2.12 \text{ kg}} \] \[ v_{\text{final}} = \frac{1.7949 \text{ kg m/s}}{2.1717 \text{ kg}} \] \[ v_{\text{final}} = 0.8267 \text{ m/s} \][/tex]
Therefore, the final velocity of the disk-stick system after the collision is approximately[tex]\( \boxed{0.83 \text{ m/s}} \)[/tex] (rounded to two decimal places).
