Question

A green block of mass m slides to the right on a frictionless floor and collides elastically with a red block of mass M which is initially at rest. After the collision the green block is at rest and the red block is moving to the right. 1)How does M compare to m?

Answer 1

Answer:  M is equal to m.

Explanation:

The question gives us two important informations:

• M is initially at rest
• m finishes at rest after collision.

In any collision, as it is asumed that no external forces can act during the collision, momentum must be conserved.

So, if we call p₁ to the momentum before collision, and p₂ to momentum after it, taking into account the information above, we can write the following:

p₁ = mv₁ + M.0 = p₂ = m.0 + Mv₂ ⇒ mv₁ = Mv₂

From the question, we also know that it was an elastic collision.

In elastic collision, added to the momentum conservation, it must be conserved the kinetic energy also.

So, if we call k₁ to the kinetic energy prior the collision, and k₂ to the one after it, we can write the following:

k₁ = 1/2 m(v₁)² + 1/2 M.0 = k₂ = 1/2m.0 + 1/2M(v₂)² ⇒ m(v₁)² = M(v₂)²

Mathematically, the only way in which both equations be true, should be with v₁ = v₂,  which is only possible if m=M too.

In this type of collision, it is said that the energy transfers from one mass to the other.

Answer 2

How M compares to m is M = m.

Law of conservation of momentum

From the law of conservation of momentum, the initial momentum, P equals the final momentum, P'

P = P'

p + p' = p" + p"' where

• p = initial momentum of green block = mv where m = mass of green block and v = initial velocity of green block,
• p = initial momentum of red block = 0 (since it is initially at rest),
• p" = final momentum of green block = 0 (since it stops) and
• p"' = final momentum of red block = MV where M = mass of red block and V = velocity of red block.

So, p + p' = p" + p"'

mv + 0 = 0 + MV

mv = MV  (1)

So, v = MV/m

Law of conservation of energy

Also, from the law of conservation of energy, the initial kinetic energy, K equals the final kinetic energy, K'

K = K'

k + k' = k" + k"' where

• k = initial kinetic energy of green block = 1/2mv²,
• k' = initial kinetic energy of red block = 0,
• k" = final kinetic energy of green block = 0 (since it stops) and
• k"' = final kinetic energy of red block = 1/2MV²

So, k + k' = k" + k"'

1/2mv² + 0 = 0 + 1/2MV²

1/2mv² = 1/2MV²

mv² = MV² (2)

Substituting v into (2), we have

mv² = MV²

m(MV/m)² = MV²

M²V²/m = MV²

M/m = 1

M = m

So, how M compares to m is M = m.

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