Answer :
Certainly! To find the final velocity of the two model cars after a perfectly inelastic collision, we can use the principle of conservation of momentum. Here's the step-by-step solution:
1. Define the variables and given data:
- Mass of Car 1, [tex]\( m_1 = 2.0 \text{ kg} \)[/tex]
- Initial velocity of Car 1, [tex]\( v_{1i} = 2 \, \text{m/s} \)[/tex]
- Mass of Car 2, [tex]\( m_2 = 1.0 \text{ kg} \)[/tex]
- Initial velocity of Car 2, [tex]\( v_{2i} = -3 \, \text{m/s} \)[/tex]
2. Understand that after a perfectly inelastic collision, the two cars stick together and move with the same final velocity, [tex]\( v_f \)[/tex].
3. Apply the conservation of momentum:
- The total initial momentum of the system is given by the sum of the momenta of both cars before the collision.
[tex]\[ p_{\text{initial}} = (m_1 \cdot v_{1i}) + (m_2 \cdot v_{2i}) \][/tex]
4. Plug in the given values:
[tex]\[ p_{\text{initial}} = (2.0 \, \text{kg} \cdot 2 \, \frac{\text{m}}{\text{s}}) + (1.0 \, \text{kg} \cdot (-3) \, \frac{\text{m}}{\text{s}}) \][/tex]
5. Calculate the total initial momentum:
[tex]\[ p_{\text{initial}} = (4 \, \text{kg} \cdot \frac{\text{m}}{\text{s}}) + (-3 \, \text{kg} \cdot \frac{\text{m}}{\text{s}}) = 1 \, \text{kg} \cdot \frac{\text{m}}{\text{s}} \][/tex]
6. Combine the masses after the collision since they stick together:
[tex]\[ m_{\text{combined}} = m_1 + m_2 = 2.0 \, \text{kg} + 1.0 \, \text{kg} = 3.0 \, \text{kg} \][/tex]
7. Set up the momentum conservation equation:
[tex]\[ p_{\text{initial}} = p_{\text{final}} \][/tex]
[tex]\[ 1 \, \text{kg} \cdot \frac{\text{m}}{\text{s}} = (m_{\text{combined}} \cdot v_f) \][/tex]
[tex]\[ 1 \, \text{kg} \cdot \frac{\text{m}}{\text{s}} = (3.0 \, \text{kg} \cdot v_f) \][/tex]
8. Solve for the final velocity [tex]\( v_f \)[/tex]:
[tex]\[ v_f = \frac{1 \, \text{kg} \cdot \frac{\text{m}}{\text{s}}}{3.0 \, \text{kg}} = 0.3333333333333333 \, \frac{\text{m}}{\text{s}} \][/tex]
9. Conclusion:
The final velocity immediately after the collision is [tex]\( 0.333 \, \frac{\text{m}}{\text{s}} \)[/tex].
So, the two cars, after colliding and sticking together, will move with a final velocity of approximately [tex]\( 0.333 \, \frac{\text{m}}{\text{s}} \)[/tex].
1. Define the variables and given data:
- Mass of Car 1, [tex]\( m_1 = 2.0 \text{ kg} \)[/tex]
- Initial velocity of Car 1, [tex]\( v_{1i} = 2 \, \text{m/s} \)[/tex]
- Mass of Car 2, [tex]\( m_2 = 1.0 \text{ kg} \)[/tex]
- Initial velocity of Car 2, [tex]\( v_{2i} = -3 \, \text{m/s} \)[/tex]
2. Understand that after a perfectly inelastic collision, the two cars stick together and move with the same final velocity, [tex]\( v_f \)[/tex].
3. Apply the conservation of momentum:
- The total initial momentum of the system is given by the sum of the momenta of both cars before the collision.
[tex]\[ p_{\text{initial}} = (m_1 \cdot v_{1i}) + (m_2 \cdot v_{2i}) \][/tex]
4. Plug in the given values:
[tex]\[ p_{\text{initial}} = (2.0 \, \text{kg} \cdot 2 \, \frac{\text{m}}{\text{s}}) + (1.0 \, \text{kg} \cdot (-3) \, \frac{\text{m}}{\text{s}}) \][/tex]
5. Calculate the total initial momentum:
[tex]\[ p_{\text{initial}} = (4 \, \text{kg} \cdot \frac{\text{m}}{\text{s}}) + (-3 \, \text{kg} \cdot \frac{\text{m}}{\text{s}}) = 1 \, \text{kg} \cdot \frac{\text{m}}{\text{s}} \][/tex]
6. Combine the masses after the collision since they stick together:
[tex]\[ m_{\text{combined}} = m_1 + m_2 = 2.0 \, \text{kg} + 1.0 \, \text{kg} = 3.0 \, \text{kg} \][/tex]
7. Set up the momentum conservation equation:
[tex]\[ p_{\text{initial}} = p_{\text{final}} \][/tex]
[tex]\[ 1 \, \text{kg} \cdot \frac{\text{m}}{\text{s}} = (m_{\text{combined}} \cdot v_f) \][/tex]
[tex]\[ 1 \, \text{kg} \cdot \frac{\text{m}}{\text{s}} = (3.0 \, \text{kg} \cdot v_f) \][/tex]
8. Solve for the final velocity [tex]\( v_f \)[/tex]:
[tex]\[ v_f = \frac{1 \, \text{kg} \cdot \frac{\text{m}}{\text{s}}}{3.0 \, \text{kg}} = 0.3333333333333333 \, \frac{\text{m}}{\text{s}} \][/tex]
9. Conclusion:
The final velocity immediately after the collision is [tex]\( 0.333 \, \frac{\text{m}}{\text{s}} \)[/tex].
So, the two cars, after colliding and sticking together, will move with a final velocity of approximately [tex]\( 0.333 \, \frac{\text{m}}{\text{s}} \)[/tex].
