Chapter 7, Problem 41P

To determine

The direction and speed of motion of the eagles after the collision

Answer to Problem 41P

Speed is $6.7\frac{\text{m}}{\text{s}}$ and the direction is $59.6°$ relative to the direction of motion of eagle A before collision

Explanation of Solution

Given:

Mass of the eagle A $=m_A=4.3\text{kg}$

Mass of the eagle B $=m_B=5.6\text{kg}$

Initial velocity of eagle A before collision $={\overrightarrow{v}}_{Ai}=v_A\widehat{i}=7.8\widehat{i}$

Initial velocity of eagle B before collision $={\overrightarrow{v}}_{Bi}=v_B\widehat{j}=10.2\widehat{j}$

Final velocity of combination of two eagles after collision $={\overrightarrow{v}}_f$

Final velocity of second ball after collision $=v_{2f}$

Direction of the velocity after collision $=\theta$

Formula Used:

Momentum is given as

  $\begin{array}{l}\overrightarrow{p}=m\overrightarrow{v}\\ \text{where }m\text{ = mass}\\ \overrightarrow{v}=\text{velocity vector}\end{array}$

According conservation of momentum principle,

  $\begin{array}{l}{\overrightarrow{p}}_i={\overrightarrow{p}}_f\\ \text{where }{\overrightarrow{p}}_i=\text{initial momentum}\\ {\overrightarrow{p}}_f=\text{final momentum}\end{array}$

Calculation:

According to conservation of momentum,

  $\begin{array}{l}\text{Total momentum of eagles before collision = Total momentum of eagles after collision}\\ {\overrightarrow{p}}_i={\overrightarrow{p}}_f\\ m_A{\overrightarrow{v}}_{Ai}+m_B{\overrightarrow{v}}_{Bi}=(m_A+m_B){\overrightarrow{v}}_f\\ (4.3)(7.8\widehat{i})+(5.6)(10.2\widehat{j})=(4.3+5.6){\overrightarrow{v}}_f\\ 33.54\widehat{i}+57.12\widehat{j}=(9.9){\overrightarrow{v}}_f\\ {\overrightarrow{v}}_f=3.4\widehat{i}+5.8\widehat{j}\end{array}$

Speed is given as

  $\left|{\overrightarrow{v}}_f\right|=\sqrt{{(3.4)}^2+{(5.8)}^2}=6.7\frac{\text{m}}{\text{s}}$

Direction is given as

  $\theta ={\tan }^{-1}\left(\frac{5.8}{3.4}\right)=59.6°$

Conclusion:

Therefore, speed is $6.7\frac{\text{m}}{\text{s}}$ and the direction is $59.6°$ relative to the direction of motion of eagle A before collision