Elastic Collisions in One Dimension — Question 10

Question 10

Mass-ratio bounds from a nearly complete speed exchange

An incident cart of mass $m$ hits a stationary cart of mass $M$ elastically. The measured signed ratio $q=v_1/u$ lies between $-0.02$ and $+0.02$. Treat this as a hard bound.

1. Find the corresponding interval for $M/m$.

2. Bound the transferred energy fraction and explain why a near-zero incident final speed is a sensitive mass-match test.

Question 10 - Solution

Mass-ratio bounds from a nearly complete speed exchange

(a) Derivation and calculation

Invert the elastic projectile relation: $$q=\frac{1-r}{1+r}\Rightarrow r=\frac{1-q}{1+q},\qquad\frac{dr}{dq}=-\frac2{(1+q)^2}<0.$$ $$\boxed{\frac{49}{51}\leq\frac{M}{m}\leq\frac{51}{49}}\quad\text{or}\quad\boxed{0.9608\leq M/m\leq1.0408}.$$

(b) Result and interpretation

The target receives the fraction not retained by the projectile: $$\eta=1-q^2,\qquad |q|\leq0.02\Rightarrow\boxed{0.9996\leq\eta\leq1}.$$ Negative $q$ means rebound from a heavier target; positive $q$ means forward motion after striking a lighter target.

Physical check

Near equal masses, the energy-transfer error is second order in the small residual speed.