Question

the figure below shows the top view of two balls colliding on a horizontal frictionless table. the path followed by the center of mass of this system is shown by the black vector. which path, labeled a, b, c, or d, will the center of mass follow after the collision? d b a c

Explanation:

Step1: Recall the law of conservation of momentum

In the absence of external forces (the table is frictionless, so no external horizontal forces), the total momentum of the system is conserved. The velocity of the center of mass \(\vec{v}_{cm}=\frac{\sum_{i = 1}^{n}m_{i}\vec{v}_{i}}{M}\), where \(M=\sum_{i = 1}^{n}m_{i}\). Since \(\vec{F}_{ext}=\frac{d\vec{P}}{dt}\) and \(\vec{F}_{ext} = 0\) (no external horizontal forces), \(\vec{P}\) (total momentum) is constant. And \(\vec{v}_{cm}=\frac{\vec{P}}{M}\), so \(\vec{v}_{cm}\) is constant.

Step2: Analyze the direction of the center - of - mass velocity

Before the collision, the center of mass has a certain velocity (given by the black vector). After the collision, since there is no external force acting on the system in the horizontal plane (frictionless table, and assuming no vertical - to - horizontal force transfer in the top - view collision), the velocity of the center of mass of the two - ball system does not change. So the path of the center of mass after the collision should be in the same direction as before the collision. Looking at the figure, path \(B\) has the same direction as the initial center - of - mass velocity (black vector).

Answer:

B.