Momentum and Kinetic Energy in Collisions — Question 7

Question 7

Maximum compression is not the final collision state

A $1\,\mathrm{kg}$ cart at $6\,\mathrm{m/s}$ encounters a stationary $2\,\mathrm{kg}$ cart through a massless ideal spring with $k=600\,\mathrm{N/m}$. The track is frictionless and the spring is initially relaxed.

1. Find their common velocity and spring compression at maximum compression.

2. Why would treating that instant as a perfectly inelastic final state be incorrect?

Question 7 - Solution

Maximum compression is not the final collision state

(a) Derivation and calculation

Maximum compression occurs when the relative speed is zero: $$V_C=\frac{1(6)+2(0)}3=2\ \mathrm{m/s}.$$ $$K_i=18\ \mathrm J,\qquad K_C=\tfrac12(3)(2)^2=6\ \mathrm J.$$ $$\tfrac12kx_{\max}^2=18-6=12\Rightarrow\boxed{x_{\max}=\sqrt{24/600}=0.20\ \mathrm m}.$$

(b) Result and interpretation

At that instant, $$v_1=v_2=V_C,\qquad K+U_s=6+12=18\ \mathrm J.$$ Equal instantaneous velocities do not mean permanent sticking. The ideal spring subsequently releases its energy and separates the objects.

Physical check

Only relative-motion energy can be stored in the internal spring.