2 10 Elastic Collision In Two Dimensions – Class 11TH Physics

2.10 Elastic Collision in Two Dimensions

Consider the motion of two balls of mass $m_{1}$ and $m_{2}$ in a straight line with velocities $v_{1}$ and $v_{2}$ respectively undergoing an elastic collision with each other as shown in Fig. 2.16.

Assume the bodies move off in different directions after collision with velocities $v_{1}'$ and $v_{2}'$ making angles $\theta_{1}$ and $\theta_{2}$ respectively with $x$ -axis.

As, the collision is elastic, so we apply both the laws of conservation of momentum and law of conservation of kinetic energy. Momentum is a vector quantity, we resolve it into its rectangular components and apply the law of conservation of momentum along both axes.

Momentum conservation along $x$ -axis is:

Momentum before collision = momentum after collision

m_{1}v_{1} + m_{2}v_{2} = m_{1}v_{1}'\cos\theta_{1} + m_{2}v_{2}'\cos\theta_{2} \quad \ldots (2.35)

Momentum conservation along $y$ -axis:

Momentum before collision = Momentum after collision

0 = m_{1}v_{1}'\sin\theta_{1} - m_{2}v_{2}'\sin\theta_{2} \quad \ldots (2.36)

Conservation of Energy

Kinetic Energy before collision = Kinetic Energy after collision

\tfrac{1}{2}m_{1}v_{1}^{2} + \tfrac{1}{2}m_{2}v_{2}^{2} = \tfrac{1}{2}m_{1}v_{1}'^{2} + \tfrac{1}{2}m_{2}v_{2}'^{2} \quad \ldots (2.37)

These equations (2.35 to 2.37) are useful in solving problems about elastic collisions in two dimensions.

2.10 Elastic Collision in Two Dimensions

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