Animated Solution for Physics - Electrostatics: A charged particle $q$ is shot towards another charged particle $Q$ which is fixed with a speed $v$. It approaches $Q$ upto a closest distance $r$ and then returns. If $q$ was given a speed $2v$, the closest distance of approach would be

Have you ever tried pushing two identical magnets together? The closer they get, the harder they push back. This is exactly what happens when a charged particle $q$ is fired towards another fixed charge $Q$ of the same sign.

Imagine the particle $q$ zooming towards $Q$ with an initial speed $v$. As it moves closer, the electrostatic force of repulsion acts like an invisible spring, slowing it down.

Eventually, the particle's speed drops to zero for a brief moment before it is pushed back. The distance between the two charges at this exact moment is called the distance of closest approach, denoted by $r$.

To solve this, we rely on one of the most powerful tools in physics: the Law of Conservation of Mechanical Energy. Since the electrostatic force is conservative, the total mechanical energy of the system remains constant.

Initially, when the particle is far away, all its energy is kinetic:

At the distance of closest approach, the particle momentarily stops. Its kinetic energy becomes zero, and all that initial energy is stored as electrostatic potential energy:

Equating the two gives us our master equation for the first case:

Now, the problem introduces a twist. What if the particle is fired with double the initial speed, $2v$? Let the new distance of closest approach be $r^{\prime }$.

We can set up a similar energy conservation equation for this second case:

To find $r^{\prime }$, we can simply divide the first equation by the second equation. This is a brilliant mathematical trick to eliminate all the constants like $m$, $k$, $Q$, and $q$.

Simplifying the left side, we get:

And simplifying the right side, we get:

Equating them:

Cross-multiplying gives us the final result:

This makes perfect physical sense! Kinetic energy depends on the square of the velocity. By doubling the speed, the initial kinetic energy became four times larger. Therefore, the particle had enough energy to push against the repulsive force and get four times closer to the fixed charge.