Question

If the speed of a particle is doubled, what happens to its kinetic energy? (a) It becomes four times larger. (b) It becomes two times larger. (c) It becomes $\sqrt{\mathbf{2}}$ times larger. (d) It is unchanged. (e) It becomes half as large.

Short answer

The correct option is (a): it becomes four times larger.

Step-by-step solution

Given information

The mass of the particle is m.

The initial speed of a particle is v.

The final speed of the particle is 2v.

Kinetic energy of a particle

The formula for the initial kinetic energy of the particle is given by the following:

$\mathbf{KE}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{mv}}^{\mathbf{2}}$

So, if the particle’s mass is higher and it travels faster, the magnitude of the kinetic energy of the particle is also higher; otherwise, it is lower.

Final kinetic energy 

According to the question, after the speed of a particle is doubled, the value of the final kinetic energy of the particle is given by the following:

$\begin{array}{l}\mathrm{KE}\text{'}=\frac{1}{2}\mathrm{m}{\left(2\mathrm{v}\right)}^2\\ \mathrm{KE}\text{'}=\frac{1}{2}\mathrm{m}\times 4{\mathrm{v}}^2\\ \mathrm{KE}\text{'}=4\times \frac{1}{2}{\mathrm{mv}}^2\\ \mathrm{KE}\text{'}=4\times \mathrm{KE}\end{array}$

From the above expression,if the speed of a particle is doubled, its kinetic energy becomes four times larger.

Hence, the correct option is (a): it becomes four times larger.