Get comprehensive solutions to your problems with IDNLearn.com. Join our interactive Q&A community and get reliable, detailed answers from experienced professionals across a variety of topics.
Sagot :
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].
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.