Animated Solution for Physics - Optics: Car B overtakes another car A at a relative speed of $40{\text{ ms}}^{-1}$. How fast will the image of car B appear to move in the mirror of focal length 10 cm fitted in car A, when the car B is 1.9 m away from the car A?

Catching the Overtaking Car

Imagine you are cruising down the highway in Car A. You glance at your convex rearview mirror and notice Car B rapidly approaching from behind. The relative speed of Car B is a brisk $40\text{ m/s}$. But here is a fascinating question: how fast does the image of Car B appear to move inside that mirror?

This isn't just a trick of the eye; it's a beautiful application of the kinematics of spherical mirrors. Let's break down the physics behind what you see.

When an object moves along the principal axis of a spherical mirror, its image also moves. The relationship between the velocity of the image ($v_{IM}$) and the velocity of the object ($v_{OM}$) is governed by the transverse magnification ($m$).

By differentiating the standard mirror formula $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$ with respect to time, we arrive at a very elegant and powerful relation:

This equation tells us two crucial things. First, the speed of the image is scaled by the square of the magnification. Second, the negative sign indicates that the image always moves in the opposite direction to the object relative to the mirror.

To use our master equation, we first need to find the magnification $m$. We could calculate the image distance $v$ first, but there is a much faster way. We can use the formula that directly relates magnification to focal length $f$ and object distance $u$:

Now, we must be extremely careful with our sign conventions. Car A's rearview mirror is convex, which means its focal length is positive. So, $f=+10\text{ cm}$. The object (Car B) is in front of the mirror, so the object distance is negative. We are given that Car B is $1.9\text{ m}$ away, which is $190\text{ cm}$. Therefore, $u=-190\text{ cm}$.

Let's substitute these values into our magnification formula:

Simplifying the denominator, we get:

So, the image is diminished to $1/20$th of the object's size. This makes sense, as convex mirrors are designed to give a wider field of view by shrinking images!

Now that we have our magnification, we can bring it back to our velocity equation. We know $m=\frac{1}{20}$ and the object velocity $v_{OM}=40\text{ m/s}$.

Squaring the magnification gives us:

The magnitude of the velocity is $0.1\text{ m/s}$, which is our final answer. But what about that negative sign?

Physically, the negative sign means that while Car B is moving towards the mirror (a positive velocity direction), its image is moving in the opposite direction. If you look into the mirror, the image appears to be moving from deep inside the mirror towards the surface (the pole).

So, the next time you check your rearview mirror, remember: the cars you see aren't just smaller; their apparent speed is drastically reduced by the square of the magnification!