Chapter 6, Problem 19P

To determine

The bullet which has the greater speed if two bullets are fired at the sametime if bullets have same kinetic energy and one bullet has twice the mass of other.

b) The one which can do the most work.

Answer to Problem 19P

Solution:a) Second bullet travels with the greatest speed with a factor of $\sqrt{2}$ .

b) Both bullets have the same Kinetic energy, so both bullets can do the same amount of work.

Explanation of Solution

Given:Kinetic energies are same for both the bullets $K.E{.}_1=K.E{.}_2$

Mass is twice of other mass $m_1=2m_2$

Formula Used:

The kinetic energy of a moving particle is given by

  $K.E=\frac{1}{2}m{v}^2$

m is the mass of the particle.

v is the velocity of the particle.

Calculations: The kinetic energy of the first bullet is

  $K.E_1=\frac{1}{2}{m}_1{v}_1{}^2$

Here, $m_1$ is mass of the firstbullet;

  $v_1$ is the speed of the first bullet.

The kinetic energy of the second bullet is

  $K.E_2=\frac{1}{2}\times m_2{v}_2{}^2$

Where $m_2$ is mass of the first bullet, $v_2$ is the speed of the secondbullet.

Since the kinetic energies are same. Equate both the kinetic energies

  $\begin{array}{l}K.E_1=K.E_2\\ \frac{1}{2}{m}_1{v}_1{}^2=\frac{1}{2}{m}_2{v}_2{}^2\end{array}$

We know that $m_1=2m_2,$ substitute this in above equation

  $\begin{array}{l}\frac{1}{2}(2m_2)v_1{}^2=\frac{1}{2}{m}_2{v}_2{}^2\\ 2v_1{}^2=v_2{}^2\end{array}$

  $\begin{array}{l}{v}_1{}^2=v_2{}^2/2\\ v_1=\sqrt{2}{v}_2\end{array}$

The second bullet travels with the greatest speed with a factor of $\sqrt{2}$

b) The kinetic energy of each bullet is converted into work. But here both the bullets have same kinetic energy, so both bullets can do the same amount of work.

Conclusion:a) Second bullet travels with the greatest speed with a factor of $\sqrt{2}$

b) Both bullets have the same Kinetic energy, so both bullets can do the same amount of work.