Question

If you want to double the kinetic energy of a gas molecule, by what factor must you increase its momentum?A) 16B) 2^(1/2)

Answer 1

Answer: $\sqrt{2}$

The linear momentum $p$ is given by the following equation:

$p=m.v$   (1)

Where $m$ is the mass and $v$ the velocity.

On the other hand, the kinetic energy $K$ is given by:

$K=\frac{p^{2}}{2m}$   (2)

Which is the same as:

$K=\frac{1}{2}m.v^{2}$

Now, if we double the kinetic energy, equation (2) changes to:

$2K=2\frac{p^{2}}{2m}$  

$2K=\frac{p^{2}}{m}$   (3)

So, if we want to obtain the kinetic energy as shown in (3), the only option that works is increasing momentum by a factor of $\sqrt{2}$ or $2^{1/2}$:

Applying this in (2):

$K=\frac{(\sqrt{2}p)^{2}}{2m}$

$K=\frac{(2p)^{2}}{2m}$

$K=\frac{p^{2}}{m}$>>>As we can see, this equation is the same as equation (3)

Therefore, the correct answer is B