Mechanics

To find the velocity of the body initially moving at 5 m/s after an elastic collision with an identical body moving at 3 m/s, we use the principles of conservation of momentum and kinetic energy, since the collision is elastic.
Step 1: Define variables
- Mass of both bodies
- Initial velocity of the first body (moving right)
- Initial velocity of the second body (moving left, opposite direction)
- Final velocities are (for the first body) and (for the second body)
Step 2: Conservation of momentum
For an elastic collision, momentum is conserved: Since : Substitute the initial velocities:
Step 3: Conservation of kinetic energy
For an elastic collision, kinetic energy is also conserved: Since , cancel and : Substitute the initial velocities:
Step 4: Solve the equations
From equation (1): Substitute into equation (2): Expand: Divide by 2: Solve this quadratic equation using the quadratic formula , where , , :
Step 5: Determine corresponding
- If : - If :
Step 6: Interpret the results
The two solutions represent the possible exchanges of velocities due to the elastic collision of identical masses. Since the bodies are identical and moving toward each other, they exchange velocities:
- The body initially at 5 m/s will move at -3 m/s (reversing direction to the left).
- The body initially at -3 m/s will move at 5 m/s (reversing direction to the right).
Step 7: Answer the question
The velocity of the body initially moving at 5 m/s after the collision is .
Step 8: Verify
- Momentum: , (conserved).
- Kinetic energy: , (conserved).
Step 9: Conclusion
The velocity is , matching option (a).