Chapter 7, Problem 64GP

To determine

To explain: Whether a player should be worry for a shot to be a scratch shot.

Answer to Problem 64GP

There is a good possibility of the shot to be a scratch shot.

Explanation of Solution

Given:

The corner pocket shot diagram is shown below.

  

Both the balls have identical mass. The colored ball is at rest initially.

Formula used:

According to the conservation of momentum principle, the total momentum before and after the collision remains constant that is,

  $\begin{array}{l}\overrightarrow{p_i}=\overrightarrow{p_f}\\ m\overrightarrow{v_a}+m\times 0=m\overrightarrow{v_a^{\text{'}}}+m\overrightarrow{v_b^{\text{'}}}\end{array}$

And, according to the conservation of kinetic energy principle, the kinetic energy of both the particles before and after the collision will remain constant that is,

  $\begin{array}{l}K{E}_i=K{E}_f\\ \frac{1}{2}m{v}_a^2+0=\frac{1}{2}m{v}_a^{\text{'}2}+\frac{1}{2}m{v}_b^{\text{'}2}\end{array}$

Calculation:

Using the momentum conservation principle, $\overrightarrow{v_a}=\overrightarrow{v_a^{\text{'}}}+\overrightarrow{v_b^{\text{'}}}$ and using kinetic energy conservation principle, $v_a^2=v_a^{\text{'}2}+v_b^{\text{'}2}$

  

Now, applying the law of cosines to the triangle of vectors,

  

  $v_a^2=v_a^{\text{'}2}+v_b^{\text{'}2}-2v_a^{\text{'}}{v}_b^{\text{'}}\cos \theta$

Comparing both of the above equations,

  $\begin{array}{l}{v}_a^{\text{'}2}+v_b^{\text{'}2}-2v_a^{\text{'}}{v}_b^{\text{'}}\cos \theta =v_a^{\text{'}2}+v_b^{\text{'}2}\\ \cos \theta =0\\ \theta ={90}^{\circ }\end{array}$

Considering the geometry from the corner pocket shot diagram, the left triangle gives

  ${\theta }_1={\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)={30}^{\circ }$ , and the right triangle gives ${\theta }_2={\tan }^{-1}\left(\frac{3}{\sqrt{3}}\right)={60}^{\circ }$ . Adding both angles while targeting the ball for the corner pocket, it makes $\theta ={90}^{\circ }$ . This appears to be a good probability of making a scratch shot.

Conclusion:

There is a good possibility of this shot to be a scratch shot.