Animated Solution for Physics - Optics: A car is fitted with a convex side-view mirror of focal length $20\text{ cm}$. A second car $2.8\text{ m}$ behind the first car is overtaking the first car at a relative speed of $15\text{ m/s}$. The speed of the image of the second car as seen in the mirror of the first one is

Imagine you are cruising down the highway, and you glance at your side-view mirror. You see a car rapidly approaching from behind. Have you ever wondered why the image in the mirror seems to move so slowly at first, but then suddenly speeds up as the car gets right next to you? This everyday phenomenon is a beautiful demonstration of the optics of convex mirrors and the calculus of motion.

In this problem, we are tasked with finding the exact speed of the image of an overtaking car. The mirror is convex with a focal length of $20\text{ cm}$ (or $0.2\text{ m}$), and the overtaking car is $2.8\text{ m}$ behind, moving at a relative speed of $15\text{ m/s}$. Let's break down the physics step-by-step.

To connect the position of the object (the real car) and its image, we start with the foundational mirror formula:

Here, $u$ is the object distance, $v$ is the image distance, and $f$ is the focal length. Because the car is moving, both $u$ and $v$ are functions of time $t$. The focal length $f$, however, is a constant property of the mirror. To find the velocities, we must differentiate this entire equation with respect to time:

Applying the chain rule, we get:

Rearranging the terms to isolate the image velocity $\frac{dv}{dt}$, we find a profound relationship:

Notice the term $\frac{v}{u}$. In optics, the lateral magnification $m$ is defined as $-\frac{v}{u}$. Therefore, the ratio $\frac{v^2}{u^2}$ is simply $m^2$. We can rewrite our velocity equation elegantly as:

This equation tells us that the longitudinal magnification (how much the velocity is scaled) is the square of the lateral magnification. The negative sign indicates that the image and object move in opposite directions relative to the coordinate system. If the object moves towards the mirror (positive velocity), the image moves towards the mirror from the other side (negative velocity).

We could find $v$ using the mirror formula and then plug it into our velocity equation. However, there is a much faster, pro-level shortcut. We can express the magnification $m$ directly in terms of the focal length $f$ and the object distance $u$:

Substituting this into our velocity equation gives us a master formula that bypasses the need to calculate $v$ entirely:

Now, let's carefully substitute our given values. We must strictly adhere to the Cartesian sign convention. The mirror is convex, so its focal length is positive: $f=+0.2\text{ m}$. The object is in front of the mirror, so its distance is negative: $u=-2.8\text{ m}$. The object is moving towards the mirror, so its velocity is positive: $v_{obj}=+15\text{ m/s}$.

Watch out for the double negative in the denominator! It becomes addition:

Simplifying the fraction inside the parenthesis:

The magnitude of the image velocity is $\frac{1}{15}\text{ m/s}$. This is incredibly slow compared to the actual car's speed of $15\text{ m/s}$! Why is this happening?

Because the car is relatively far away ($2.8\text{ m}$) compared to the focal length ($0.2\text{ m}$), the lateral magnification $m$ is very small ($\frac{1}{15}$). Since the velocity scales with $m^2$, the image velocity is scaled down by a massive factor of $225$.

However, as the car gets closer, $u$ approaches $0$, and the magnification $m$ approaches $1$. This means $m^2$ also approaches $1$. Therefore, as the car gets right up to your bumper, its image will suddenly appear to accelerate and match the real car's speed. This non-linear perceived acceleration is exactly why side-view mirrors carry the warning: "Objects in mirror are closer than they appear."