To explain how momentum is conserved after collision, consider the simplified case of a one-dimensional collision between two objects:
Case 1: Elastic Collision between Two Moving Objects
- Before collision: Object 1 with mass m1 and velocity u1, object 2 with mass m2 and velocity u2.
- During collision: The collision is assumed to be elastic, meaning no loss of kinetic energy. The forces involved in the collision are conservative and do not change the total momentum of the system.
- After collision: Object 1 with mass m1 and velocity v1, object 2 with mass m2 and velocity v2.
By applying the principle of conservation of momentum, we have:
```
Total initial momentum = Total final momentum
m1u1 + m2u2 = m1v1 + m2v2
```
In this case, since the collision is elastic, the relative velocities before and after the collision satisfy:
```
(v1 - u1) = (v2 - u2)
```
By rearranging the equation, we can see that the relative motion between the objects remains unchanged after the collision, ensuring conservation of momentum.
Case 2: Inelastic Collision Leading to Sticking Together
Consider another scenario where the collision between the two objects is inelastic. After the collision, the objects stick together and move as a composite object.
- Before collision: Object 1 with mass m1 and velocity u1, object 2 with mass m2 and velocity u2.
- After collision: Combined object with mass (m1 + m2) and velocity v.
Again, applying the conservation of momentum:
```
Total initial momentum = Total final momentum
m1u1 + m2u2 = (m1 + m2)v
```
Solving for v, we find the velocity of the combined object after the collision:
```
v = (m1u1 + m2u2) / (m1 + m2)
```
In this case, the final velocity of the combined object is the weighted average of the initial velocities, taking into account the different masses of the objects.
These examples illustrate how momentum is conserved in collisions, whether elastic or inelastic. The principle ensures that the total momentum of a closed system remains unchanged, regardless of the forces acting within the system.
