What change to the report’s data would give a correct verification of momentum conservation?

Question

The following report for a collision of two marbles has an error. Marble 1 has a mass of 3 g and an initial speed of 6 m/s right. After the collision it has a final speed of 4 m/s to the right. Marble 2 has a mass of 2 g, and is initially at rest. After the collision it is moving at a final speed of 4 m/s to the right. What change to the report’s data would give a correct verification of momentum conservation?

Answer 1

Let

m₁ = 3g be the mass of marble 1,

m₂ = 2g be the mass of marble 2,

v₁ = 6 m/s be the initial speed of marble 1,

v₂ = 0 m/s be the initial speed of marble 2,

w₁ = 4 m/s be the final speed of marble 1,

w₂ = 4 m/s be the final speed of marble 2.

The final momentum is directed to the right and has magnitude

m₁w₁ + m₂w₂ = 3×4 + 2×4 = 20 gm/s

For conservation of momentum the initial momentum must also be to the right.

So one improvement in the report would be to say that the initial velocity is to the right.

The magnitude of the initial momentum is

m₁v₁ + m₂v₂ = 3×6 + 2×0 = 18 gm/s

There are a many changes to the report that would make the two momenta equal.

One thing we should not change is w₁ = w₂.

That indicates that the marbles stick together, which implies inelastic collision,

which then does require conservation of mechanical energy.

Here are some examples of a change to the report that would make the two momenta equal:

1) v₂ = 1 m/s

2) w₁ = w₂ = 18/5 m/s

3) v₁ = 20/3 m/s

4) m₂ = 1.5 g

And there are many other possibilities.