Скачать или смотреть Solve elastic collisions using the center of mass frame trick!

Two carts initially head toward each other and collide in a perfectly elastic one-dimensional collision, meaning both momentum and energy are conserved.

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This time, we approach the problem by switching to the center of mass reference frame. First, we find the center of mass velocity of the system by using a theorem of classical mechanics: p_net=Mv_cm. We solve for v_cm and calculate the net momentum by adding up the momenta of the carts in the initial state.

After we find the center of mass velocity of the carts, we transform our initial state into the center of mass reference frame by thinking about how the velocities of the carts should differ from the perspective of an observer moving along with the center of mass of the system.

We intend to exploit the same theorem of mechanics one more time: the net momentum is zero in the center of mass reference frame, and this zero is critically important: knowing that the net momentum is zero in the center of mass reference frame will ultimately allow us to solve the elastic collision by guessing the final velocities instead of doing a ton of algebra!

So in the center of mass frame, the kinetic energy is conserved and the momentum is conserved (zero in both the initial and final states). This means that a simple reflection of the velocities will conserve both energy and momentum!

Finally, we have to take our final velocities in the center of mass frame and transform back to the lab frame. We adjust both of the final velocities according to what they should look like when we stop moving along with the center of mass, and we've got the final solution of the 1D elastic collision without doing any algebra!