Collisions

Question

A particle of mass m moving with speed v collides elastically with another particle of mass 2m at rest. The fraction of energy lost by the colliding particle is:

RankGuru · Accepted Answer

1. **Elastic Collision Formula**: Final velocity $v_1 = \left(\frac{m_1 - m_2}{m_1 + m_2}\right)u_1 + \left(\frac{2m_2}{m_1 + m_2}\right)u_2$.
2. **Given Values**: $m_1 = m$, $m_2 = 2m$, $u_1 = v$, $u_2 = 0$.
   $$v_1 = \left(\frac{m - 2m}{m + 2m}\right)v = -\frac{v}{3}$$
3. **Fractional Energy Calculation**:
   - Initial Kinetic Energy $(K_i) = \frac{1}{2}mv^2$.
   - Final Kinetic Energy $(K_f) = \frac{1}{2}m\left(-\frac{v}{3}\right)^2 = \frac{1}{2}m\left(\frac{v^2}{9}\right) = \frac{K_i}{9}$.
4. **Energy Lost**: $\Delta K = K_i - K_f = K_i - \frac{K_i}{9} = \frac{8}{9}K_i$.
5. **Fraction lost**: $\frac{\Delta K}{K_i} = \frac{8}{9}$.

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