Animated Solution for Physics - Kinematics: Two cars are moving at constant speeds; one on a circular path of radius $R=200\text{ m}$ and the other on a straight road. Magnitude $v$ of velocity of one car relative to the other has been recorded at regular intervals of time and data thus obtained is represented in a graph as shown in the figure. Calculate speeds of both the cars relative to the ground.

Imagine two cars in motion. One is cruising effortlessly along a perfectly straight highway, while the other is navigating a circular track with a radius of $R=200\text{ m}$. Both are moving at constant speeds. We are not given their individual speeds, nor their exact positions. Instead, we are handed a rather abstract piece of information: a graph plotting the magnitude of their relative velocity, $v=|{\vec{v}}_{rel}|$, against time.

At first glance, extracting the individual speeds from this single curve seems like magic. But physics is all about finding the hidden connections between geometry and motion.

The most glaring feature of the provided graph is the sharp dip at $t=25\text{ s}$, where the relative speed touches exactly zero. What does a relative speed of zero mean physically?

Relative velocity is defined as the vector difference: ${\vec{v}}_{rel}={\vec{v}}_1-{\vec{v}}_2$. For the magnitude of this vector to be zero, the vector itself must be the null vector. This implies that at exactly $t=25\text{ s}$, ${\vec{v}}_1={\vec{v}}_2$.

This is a massive revelation! It tells us that at this specific instant, both cars are moving in the exact same direction and, crucially, with the exact same speed. Let's denote this common, constant speed as $u$.

To dig deeper, we need to understand how the relative speed changes over time. We start with the fundamental dot product relation for the square of the magnitude:

Differentiating both sides with respect to time using the chain rule, we get:

Since the derivative of velocity is acceleration, $\frac{d{\vec{v}}_{rel}}{dt}={\vec{a}}_{rel}$. Rearranging the terms gives us a beautiful expression for the rate of change of relative speed:

Notice that the term $\frac{{\vec{v}}_{rel}}{v}$ is simply the unit vector ${\hat{v}}_{rel}$ pointing in the direction of the relative velocity.

Let's analyze the relative acceleration, ${\vec{a}}_{rel}={\vec{a}}_1-{\vec{a}}_2$.

The car on the straight road is moving with a constant velocity (constant speed and constant direction), which means its acceleration is zero (${\vec{a}}_2=0$).

The car on the circular track is moving with a constant speed $u$, but its direction is continuously changing. This means it experiences a purely centripetal acceleration directed towards the center of the circle, with a magnitude of $|{\vec{a}}_1|=\frac{u^2}{R}$.

Therefore, the relative acceleration is simply the centripetal acceleration of the first car: ${\vec{a}}_{rel}={\vec{a}}_1$.

Now, let's look at the instant right after $t=25\text{ s}$. The relative velocity grows from zero exactly in the direction of the relative acceleration. This means the dot product ${\hat{v}}_{rel}\cdot {\vec{a}}_{rel}$ simplifies to just the magnitude of the acceleration.

Thus, the slope of the speed-time graph right after the minimum point is:

We now turn our attention back to the graph. The curve forms a sharp 'V' shape at $t=25\text{ s}$. If we draw a tangent line to the curve immediately after this point (for $t>25$), we can determine its slope by identifying two points it passes through.

Observing the grid carefully, the tangent line originating from $(25,0)$ passes perfectly through the intersection at $(35,20)$.

Calculating the slope (rise over run):

We have successfully bridged the theoretical physics with the graphical data. We know the theoretical slope is $\frac{u^2}{R}$, and the measured slope is $2{\text{ m/s}}^2$. Equating them:

Substitute the given radius $R=200\text{ m}$:

Since we established earlier that both cars share this common speed $u$, we conclude that the speed of each car relative to the ground is $20\text{ m/s}$. A truly elegant problem that seamlessly weaves together vector kinematics, calculus, and graphical analysis!