Two objects of mass m1 = m and m2 = 4m undergo a completely inelastic collision in one dimension. If the two objects are at rest after the collision, what was the ratio of their speeds before the collision?

v1/v2=

To find the ratio of the speeds before the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.

Before the collision, both objects are at rest, so their initial momentum is zero. After the collision, the objects stick together and move with a common final velocity.

Let's denote the velocity of the combined objects after the collision as V_f.

The momentum of the first object before the collision is given by:
p1_initial = m1 * v1_initial = m * v1_initial

The momentum of the second object before the collision is given by:
p2_initial = m2 * v2_initial = 4m * v2_initial

Since both objects are at rest initially, the total initial momentum is:
p_initial = p1_initial + p2_initial = 0 + 0 = 0

After the collision, the combined object will have a mass of m1 + m2 = m + 4m = 5m.

Therefore, the momentum of the combined object after the collision is:
p_final = (m1 + m2) * V_f = 5m * V_f

Since momentum is conserved, we have:
p_initial = p_final
0 = 5m * V_f

Since V_f is not zero (as the objects are at rest after the collision), we can conclude that the total initial momentum is zero.

Now we can find the ratio of the speeds before the collision:

v1_initial / v2_initial = (p1_initial / m1) / (p2_initial / m2)
v1_initial / v2_initial = (m * v1_initial / m) / (4m * v2_initial / 4m)
v1_initial / v2_initial = (v1_initial) / (4 * v2_initial)

To find the ratio, we need to divide both sides of the equation by v1_initial:
v1_initial / v1_initial = (v1_initial / (4 * v2_initial)) / (v1_initial / v1_initial)
1 = (v1_initial / (4 * v2_initial)) / 1
1 = (v1_initial / (4 * v2_initial))

Therefore, the ratio of the speeds before the collision is:
v1_initial / v2_initial = 1 / 4

Thus, v1_initial / v2_initial = 1/4.