Problem 1

A cart of mass

m

moves at speed

u>0

along a frictionless straight track and undergoes a one-dimensional elastic collision with a stationary cart of mass

2m

. Find the signed velocity of each cart after the collision, taking the first cart's initial direction as positive. Treat the carts as point masses and neglect external horizontal impulse during contact. (a) Write momentum conservation. (b) Use the elastic-collision condition to obtain a second relation and calculate both signed velocities. (c) Check the kinetic-energy balance and interpret the signs without confusing velocity with speed.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\frac12 m u^2=\frac12mv_1^2+\frac12(2m)v_2^2\quad\Rightarrow\quad u^2=v_1^2+2v_2^2

problems.proof.analysis

Because the collision is elastic, total kinetic energy is unchanged. Canceling the common factor $m/2$ gives the second equation.

problems.proof.pitfall. Keep the sign convention fixed and distinguish signed quantities from magnitudes.