Consider a collision in one dimension that involves two objects of masses of 4.5 kg and 6.5 kg. The smaller Mass has an initial velocity of 12m/s and the larger mass is initially at rest. The final velocity of the smaller mass is 8.0m/s and calculate the final velocity of the larger object.

If an object of mass [tex]m[/tex] is travelling at a velocity of [tex]v[/tex], the momentum [tex]p[/tex] of that object would be [tex]p = m\, v[/tex].

Momentum before collision:

[tex]4.5\; {\rm kg}[/tex] mass: [tex](4.5\; {\rm kg})\, (12\; {\rm m\cdot s^{-1}}) = 54\; {\rm kg \cdot m\cdot s^{-1}}[/tex].
[tex]6.5\; {\rm kg}[/tex] mass: [tex]0\; {\rm kg \cdot m \cdot s^{-1}}[/tex] (since [tex]v = 0\; {\rm m\cdot s^{-1}}[/tex].)

Total momentum before collision:

Momentum after collision:

[tex]4.5\; {\rm kg}[/tex] mass: [tex](4.5\; {\rm kg})\, (8.0\; {\rm m\cdot s^{-1}}) = 36\; {\rm kg \cdot m\cdot s^{-1}}[/tex].
[tex]6.5\; {\rm kg}[/tex] mass: needs to be found.

Momentum is conserved immediately before and after collisions. Hence, the total momentum of the two masses after the collision would be equal to the sum of their momentum before the collision, [tex]54\; {\rm kg \cdot m\cdot s^{-1}}[/tex].

Subtract the momentum of the [tex]4.5\; {\rm kg}[/tex] mass after the collision from the sum of momentum after the collision: [tex]54\; {\rm kg \cdot m\cdot s^{-1}} - 36\; {\rm kg \cdot m \cdot s^{-1}} = 18\; {\rm kg \cdot m \cdot s^{-1}}[/tex].

Hence, the momentum of the [tex]6.5\; {\rm kg}[/tex] mass after the collision would be [tex]18\; {\rm kg \cdot m \cdot s^{-1}}[/tex]. The velocity of this mass would be:

[tex]\begin{aligned}v &= \frac{p}{m} \\ &= \frac{18\; {\rm kg\cdot m \cdot s^{-1}}}{6.5\; {\rm kg}} \approx 2.8\; {\rm m\cdot s^{-1}}\end{aligned}[/tex].