Chapter 6, Problem 16P

(a) If the kinetic energy of a particle is tripled, by what factor has its speed increased? (b) If the speed of a particle is halved, by what factor does its kinetic energy change?

To determine

a)The amount of speed increased if the kinetic energy of a particle is tripled.

(b) The amount of change in kinetic energy if the speed of a particle is halved.

Answer to Problem 16P

Solution:a) The final speed becomes $\sqrt{3}$ times the initial speed when kinetic energy is tripled.

b) When speed is halved the final kinetic energy is quarter of the initial kinetic energy.

Explanation of Solution

Given:a) The kinetic energy of a particle is tripled

b) The speed of a particle is halved

Formula Used:

The kinetic energy of a moving particle is obtained by

  $K.E=\frac{1}{2}m{v}^2$

m is the mass of the particle

v is the velocity of the particle

Calculation:

a) When the kinetic energy of a particle is tripled:

Initial velocity be $v_1$ , which is given by

  $v_1=\sqrt{\frac{2K.E.}{m}}$

When kinetic energy of a particle is tripled put $K.E.=3K.E.$

  $\begin{array}{l}{v}_2=\sqrt{\frac{2K.E.}{m}}\\ v_2=\sqrt{\frac{2\times 3K.E.}{m}}\end{array}$

The ratio of final to initial speed is

  $\begin{array}{l}\frac{v_2}{v_1}=\frac{\sqrt{\frac{2\times 3K.E}{m}}}{\sqrt{\frac{2K.E}{m}}}\\ \frac{v_2}{v_1}=\sqrt{3}\\ v_2=\sqrt{3}{v}_1\end{array}$

Therefore, the final speed becomes $\sqrt{3}$ times the initial speed when kinetic energy is tripled.

b) When the speed of a particle is halved

  $K.E_1=\frac{1}{2}m{v}^2$ is the initial kinetic energy.

When its speed is halved $v_2=\frac{v}{2}$ , then kinetic energy will be

  $\begin{array}{l}K.E_2=\frac{1}{2}m{\left(\frac{v}{2}\right)}^2\\ K.E_2=\frac{1}{2}\times \frac{1}{4}mv\end{array}$

  $K.E_2=\frac{1}{4}K.E_1$

When speed is halved the final kinetic energy is quarter of the initial kinetic energy.

Conclusion:a) The final speed becomes $\sqrt{3}$ times the initial speed when kinetic energy is tripled.

b) When speed is halved the final kinetic energy is quarter of the initial kinetic energy.