Chapter 7, Problem 43P

To determine

The angle between initial directions of an object.

Answer to Problem 43P

The angle between the initial directions is $2\theta =141°$ .

Explanation of Solution

Given:

The mass of the object 1 is $m_1=m$ .

The mass of object 2 is $m_2=m$ .

The initial velocity of objects is v.

The final velocity of objects is $\frac{v}{3}$ .

Formula used:

The expression for law of conservation of momentum is,

  $m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$

Here, $m_1$ is the mass of the object 1, $m_2$ is the mass of object 2, $u_1$ is the initial velocity of object 1, $u_2$ is the initial velocity of the object 2, $v_1$ is the final velocity of object 1, and $v_2$ is the final velocity of the object 2.

Calculation:

Sketch the motion of objects before and after collision as shown below.

  

The angle between the objects initial direction by using the relation is,

  $\begin{array}{l}{m}_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2\\ mv\cos \theta +mv\cos \theta =m\frac{v}{3}+m\frac{v}{3}\\ 2mv\cos \theta =\frac{2mv}{3}\\ \cos \theta =\frac{1}{3}\\ \theta =70.5°\end{array}$

The angle between the objects initial direction is $2\theta$ .

The angle between the objects initial direction is,

  $\begin{array}{l}2\theta =2\times 70.5°\\ 2\theta =141°\end{array}$

Conclusion:

Thus, the angle between the objects initial direction is $2\theta =141°$ .