Answer:
v₁ = 0.45 m/s or v₂ = -0.10 m/s
Step-by-step 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.