Momentum and Kinetic Energy in Collisions — Question 10

Question 10

Uncertainty may prevent a confident elastic label

An isolated two-body experiment bounds the initial kinetic energy by $[19.8,20.2]\,\mathrm J$ and the final by $[19.6,20.0]\,\mathrm J$. Treat the intervals as independent hard bounds.

1. Find the full interval for $K_i-K_f$.

2. Can the measurements prove an inelastic collision, or are elastic and superelastic outcomes still compatible?

Question 10 - Solution

Uncertainty may prevent a confident elastic label

(a) Derivation and calculation

For independent bounded intervals, the smallest and largest losses use opposite endpoints: $$L_{\min}=19.8-20.0=-0.2\ \mathrm J,\qquad L_{\max}=20.2-19.6=0.6\ \mathrm J.$$ $$\boxed{L=K_i-K_f\in[-0.2,0.6]\ \mathrm J}.$$

(b) Result and interpretation

The interval contains each sign: $$L>0:\text{ inelastic},\qquad L=0:\text{ elastic},\qquad L<0:\text{ superelastic}.$$ None of these classifications can be excluded by the supplied uncertainty bounds.

Physical check

A positive difference of central values is not sufficient to establish a strictly positive loss.