Worked exercise — elastic collision in 1D

Problem

A body ($m_1 = 3\;\text{kg}$, $v_1 = 25\;\text{m/s}$) collides with a body at rest ($m_2 = 5\;\text{kg}$, $v_2 = 0$). The collision is elastic. Find the final velocities.

Solution

Elastic-collision trick (the relative velocity reverses): $v_1 - v_2 = v_2' - v_1'$, i.e. $25 - 0 = v_2' - v_1' \quad\Rightarrow\quad v_2' = v_1' + 25$

Conservation of $\vv{p}$: $m_1 v_1 = m_1 v_1' + m_2 v_2' \quad\Rightarrow\quad 3\cdot 25 = 3v_1' + 5v_2'$ Substituting $v_2' = v_1' + 25$: $75 = 3v_1' + 5(v_1' + 25) = 8v_1' + 125 \quad\Rightarrow\quad v_1' = -\frac{50}{8} = -6{,}25\;\text{m/s}$ $v_2' = v_1' + 25 = -6{,}25 + 25 = 18{,}75\;\text{m/s}$ $\ev{v_1' = -6{,}25\;\mathrm{m/s},\quad v_2' = 18{,}75\;\mathrm{m/s}}$ Body 1 (lighter) bounces backwards (negative velocity), body 2 (heavier) moves forward. Kinetic energy check: before $\tfrac{1}{2}\cdot 3\cdot 25^2 = 937{,}5\;\text{J}$; after $\tfrac{1}{2}\cdot 3\cdot 6{,}25^2 + \tfrac{1}{2}\cdot 5\cdot 18{,}75^2 = 58{,}6 + 878{,}9 = 937{,}5\;\text{J}$: it is conserved, as it must be.

Links

Topics: Momentum and collisions Concepts: Elastic collision · Conservation of momentum · Kinetic energy Skills: Symbolic setup