Reading a CM position graph

Problem

Graph reading: $x_\text{cm}(t)$. In an $x$ vs $t$ graph are plotted the positions of two particles that collide elastically: one starts from the top left with slope $+5\;\text{m/s}$, the other from the bottom right with slope $-3\;\text{m/s}$. Sketch $x_\text{cm}(t)$ and describe how it behaves during and after the collision.

Solution

The system is isolated: no external forces, so $\vec v_\text{cm}$ is constant and $x_\text{cm}(t)$ is a single straight line, with slope equal to $v_\text{cm}$. The two particles change velocity in the collision (their lines “break” at the collision point), but the centre of mass doesn’t notice: $\ev{x_\text{cm}(t) = \text{straight line of constant slope } v_\text{cm}}$ During the collision, the CM line undergoes no change in slope: the collision, being due to internal forces, leaves no trace on the motion of the barycentre. The red line crosses the collision point without breaking.

Links

Topics: Centro di massa Concepts: Centro di massa · Conservazione della quantità di moto Skills: Lettura dei grafici