A billiard ball of mass m_1, moves with speed, v_0, towards two billiard balls a

Question

A billiard ball of mass m_1, moves with speed, v_0, towards two billiard balls at rest, each with mass, m_2, as shown in the figure. Calculate the position and velocity of the center of mass when m_1 is a distance d from the center of the two m_2 (be sure to include direction). Take the coordinate system to be at the midpoint between masses, m_2, as shown. The masses collide in such a way that mass m_1 comes to rest, while masses m_2 each move off with speed, V_2, at an angle, theta, to the horizontal. If mi = m_2, calculate the magnitude of V_2 and the angle, theta. Assume the collision to be a perfectly elastic collision. What is the velocity of the center of mass after the collision?

Explanation / Answer

As the collision is elastic, the momentum and the kinetic energy of the system will remain conserved. We will use this to solve the given problems.

Part a.) Let us assume that the origin is situated at the centre of mass m1, so the X coordinate of COM of the system would be given as:

X = 2m(d) + m(0) / 3m = 2d/3

Therefore the centre of mass of the system would be at 2d/3 from m1 or d/3 from the two masses.

Also, for the system to have the same linear momentum as m1, we must have:

m1v = 3m1V

or, V = v/3 is the required velocity of the system's COM.

Part b.) Now, after collision the momentum and kinetic energy must remain same. As both the masses on the right move with the same velocity afterwards and are of same mass as m1 we can write:

0.5mv2 = 0.5mV2 + 0.5mV2

or, V = v/2 is the speed of each of the masses on the right.

Also, for conservation of linear momentum, we can write:

mv = 2mv2Cos or,

Cos = 1/2

Therefore the angle of motion of each of the masses is 45 degrees with the horizontal.

Part C.) The centre of mass suffers no external force in the process, hence the net linear momentum must remain conserved and for that to happen, it will continue to move along the same direction with the same velocity, that is, v/3

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