Two objects of mass 4m and 6m undergo a completely inelastic collision (they stick together) in one dimension. If the two objects are at rest after the collision, what was the ratio of their speeds before the collision?

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BlackTom AI · Accepted Answer

The problem involves a completely inelastic collision in one dimension where the two objects stick together after impact. The key principle here is conservation of linear momentum, since external impulses are neglected.

First, record the given masses and their states: object A has mass 4m, object B has mass 6m. After the collision they both move with the same final velocity v_f (because they stick together), but the problem states they are at rest after the collision, so v_f = 0.

Before the collision, the total momentum must be zero if the final momentum is zero (since momentum is conserved and the final momentum is 0 when v_f = 0). Therefore:

(4m) * v_A + (6m) * v_B = 0

The masses share a common factor m, which cancels out:

4 v_A + 6 v_B = 0

This relation can be rearranged to relate the speeds:

4 v_A = -6 v_B  =>  v_A = - (6/4) v_B = - (3/2) v_B

The speeds are magnitudes, so |v_A| = (3/2) |v_B|. The ratio of their speeds before the collision is therefore:

|v_A| / |v_B| = 3/2.

Now, evaluate the answer choices:

- Option 1: 2. This would imply |v_A| = 2 |v_B|, which contradicts the derived linear relation 4 v_A + 6 v_B = 0.
- Option 2: 3/2. This matches the calculation |v_A| = (3/2) |v_B|, consistent with momentum conservation and the final rest condition.
- Option 3: 4/3. This would require |v_A| to be (4/3) |v_B|, which is not supported by the equation 4 v_A + 6 v_B = 0.
- Option 4: 1/3. This would imply a much smaller speed for object A relative to B, again inconsistent with the momentum balance.
- Option 5: 1/4. This is also inconsistent with the momentum relation derived from setting the final momentum to zero.

In summary, the only ratio that satisfies the momentum conservation condition with v_f = 0 is |v_A| : |v_B| = 3 : 2, i.e., 3/2. The underlying reason is that a heavier body (6m) must carry proportionally more momentum to balance the lighter body (4m) when the combined system ends at rest, leading to v_A = -(3/2) v_B and thus the speed ratio 3/2.

2025 Fall PHYS 2205 TR 11:00 - 12:15 _ 89122 PHYS 2205 Exam 2- Requires Respondus LockDown Browser

Two objects of mass 4m and 6m undergo a completely inelastic collision (they stick together) in one dimension. If the two objects are at rest after the collision, what was the ratio of their speeds before the collision?

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