If you want to double the kinetic energy of a gas molecule,by what factor must you increase its momentum?
A) $\sqrt { 2 }$
B)2
C)2 $\sqrt { 2 }$
D)4
E)16

Question

Quizplus · Accepted Answer

The kinetic energy of a particle is given by $KE=\frac{1}{2}mv^2$, where m is the mass of the particle and v is its velocity. The momentum of a particle is given by $p=mv$. We want to double the kinetic energy of the particle, which can be achieved by increasing the velocity of the particle. Let the initial velocity of the particle be v1, then its initial momentum is given by p1 = mv1. If we want to double the kinetic energy of the particle, its final velocity (v2) should be such that its final kinetic energy is given by $\frac{1}{2}m{v_{2}}^{2}=2*\frac{1}{2}m{v_{1}}^{2}=m{v_{1}}^{2} $. Solving for v2, we get $v_{2}=\sqrt {2}v_{1}$. Therefore, the momentum of the particle after doubling its kinetic energy will be $p_{2}=mv_{2}=m\sqrt {2}v_{1}$. The factor by which the momentum has increased can be calculated as $p_{2}/p_{1}=\frac{m\sqrt {2}v_{1}}{mv_{1}}=\sqrt {2}$. Therefore, to double the kinetic energy of a particle, we need to increase its momentum by a factor of $\sqrt {2 }$.

If You Want to Double the Kinetic Energy of a Gas