Could some explain this question to me in detail? it is extremely vital for me to understand this question. in advance!

question is in the image

v₁ = 0.45 m/s or v₂ = -0.10 m/s

Explanation:

Marble 1 has a mass of m and an initial velocity of -0.30 m/s.

Marble 2 has a mass of m/2 and an initial velocity of 0.80 m/s.

Momentum is conserved, so:

m (-0.30) + m/2 (0.80) = m v₁ + m/2 v₂

-0.3m + 0.4m = m v₁ + m/2 v₂

0.1m = m v₁ + m/2 v₂

0.1 = v₁ + 1/2 v₂

One of the marbles has half the speed after the collision.  Let's first assume that it's Marble 1.  That means v₁ = -0.15 m/s or 0.15 m/s.

If v₁ = -0.15 m/s:

0.1 = -0.15 + 1/2 v₂

0.25 = 1/2 v₂

0.5 = v₂

This isn't possible; at least one of the marbles must change direction.

If v₁ = 0.15 m/s:

0.1 = 0.15 + 1/2 v₂

-0.05 = 1/2 v₂

-0.1 = v₂

This is a possible solution.

Now, let's assume that Marble 2 is the one that moves at half the speed.  That means v₂ = -0.40 m/s or 0.40 m/s.

If v₂ = -0.40 m/s:

0.1 = v₁ + 1/2 (-0.40)

0.25 = v₁ − 0.2

0.45 = v₁

This is a possible solution.

If v₂ = 0.40 m/s:

0.1 = v₁ + 1/2 (0.40)

0.1 = v₁ + 0.2

-0.1 = v₁

This isn't possible; at least one of the marbles must change direction.

Therefore, there are two possible answers: v₁ = 0.45 m/s, or v₂ = -0.10 m/s.

option number c is the right answer.