2014: Problem 16

Topic: Collisions
Concepts: 1D elastic collision, 1D inelastic collision, Limiting cases

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Solution:

We study the two limiting cases of a perfectly inelastic collision and an elastic collision which will provide us with the bounds that apply to any generic collision.

For a perfectly inelastic collision, the masses stick together so

$$m_1v_0=(m_1+m_2)v_1$$

$$r_1=\frac{v_1}{v_0}=\frac{m_1}{m_1+m_2}=\frac{1}{1+\alpha}$$

For an elastic collision, recall

$$v_1=\frac{m_1-m_2}{m_1+m_2}v_0$$

$$r_1=\frac{v_1}{v_0}=\frac{m_1-m_2}{m_1+m_2}=\frac{1-\alpha}{1+\alpha}$$

Thus,

$$\frac{1-\alpha}{1+\alpha} \leq r_1 \leq \frac{1}{1+\alpha}$$

so the answer is B.

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