Physic Labs

Problem 1

A cart of mass mm moves at speed u>0u>0 along a frictionless straight track and undergoes a one-dimensional elastic collision with a stationary cart of mass 2m2m. 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.
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u2−v12=2v22⇒(u−v1)(u+v1)=v2(u−v1)u^2-v_1^2=2v_2^2\quad\Rightarrow\quad (u-v_1)(u+v_1)=v_2(u-v_1)
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Use 2v2=u−v12v_2=u-v_1 from momentum conservation to replace 2v222v_2^2 by v2(u−v1)v_2(u-v_1). Moving terms and factoring gives (u−v1)(u+v1−v2)=0(u-v_1)(u+v_1-v_2)=0.