Conservation of Momentum, which follows from the fact that the objects exert equal and opposite forces on one another for equal times,  assures us that the momentum changes of the objects are equal and opposite.

We can find the momentum change of the first object:

• The first object has a change in velocity of -1 m/s - 4 m/s = -5 m/s.
• Its change in momentum is therefore

• `dp1 = m1 `dv1 = 6 kg * -5 m/s = -30 kg m/s.

The momentum change of the second object is unknown, but its mass is known.  Its momentum change being the product of its mass and its change in velocity, it is easy to find its velocity change:

• The final velocity v2' of the second object is easily found from its initial velocity and its change in velocity:

• v2' = v02 + `dv2 = -5 m/s + 3.75 m/s = -1.25 m/s.

We could have solved this problem from the more detailed statement of momentum conservation for two objects of constant mass:

We are given m1, v1, v1', m2 and v2.  The only thing we don't know is v2'.

Knowing that in a system of two objects, in which the only forces acting are the forces exerted on the objects by one another, the forces will be equal and opposite and therefore the momentum changes will be equal and opposite, we have the statement `dp2 = - `dp1.   In terms of masses and velocity changes this tells us that

• m2 `dv2 = m1 `dv1

In the present situation, where we know v2 but don't know `dv2, we can solve this equation for `dv2, obtaining

Knowing `dv2 we can find the final velocity v2' = v2 + `dv2 of the second object.

Alternatively we can simply solve the conservation of momentum equation

for v2', obtaining

• v2' = ( m1 v1 - m1 v1' + m2 v2 ) / m2.