Two cars A and B simultaneously start a race. Velocity v of the car A varies with time t according t

Q: Why can't I just find the equation of the curve?

You could try, but the problem doesn't give you enough data to find a unique equation. And even if you assumed a parabola, the math would be a nightmare. The beauty of this problem is that the exact shape of the curve *doesn't matter*!

Q: What if car B didn't have zero acceleration, but just a lower acceleration?

Then the horizontal time gap wouldn't be a constant 1 second. You would need to know exactly how the time gap varies with velocity to perform the horizontal integration.

Q: Why is the area between the curves equal to the difference in distance?

Because distance is the integral of velocity with respect to time. $\Delta s = \int v_A dt - \int v_B dt = \int (v_A - v_B) dt$. Geometrically, the integral of the difference of two functions is exactly the area enclosed between their graphs.

Q: I don't get the horizontal integration. How does $\int t dv$ give distance?

It doesn't give distance directly! The area of the region is $\int (v_A - v_B) dt$. By changing the perspective, the *same geometric area* can be calculated as $\int (t_B - t_A) dv$. We are just calculating the area of a shape in a smarter way.

Q: Why did we integrate from 20 to 50?

Because below $20\text{ m/s}$, the cars have the exact same velocity profile, so the area between them is zero. Above $50\text{ m/s}$, both cars are moving at a constant $50\text{ m/s}$, so again, the area between them is zero. The only region where their graphs differ is between $v = 20$ and $v = 50$.

Question

Two cars A and B simultaneously start a race. Velocity $v$ of the car A varies with time $t$ according to the graph shown in the figure. It acquires a velocity $50	ext{ m/s}$ few seconds before $t = 100	ext{ s}$ and thereafter moves with this speed. Car B runs together with car A till both acquire a velocity $20	ext{ m/s}$, after this car B moves with zero acceleration for one second and then follows velocity-time profile identical to that of A with a delay of one second. In this way, car B acquires the velocity $50	ext{ m/s}$ one second after A acquires it. How much more distance $\Delta s$ does the car A cover in the first $100	ext{ s}$ as compared to the car B?

Options:

(A) $\Delta s = 30	ext{ m}$

(B) $\Delta s < 30	ext{ m}$

(C) $\Delta s = 20	ext{ m}$

(D) Insufficient information.

Sigma Percentile · Accepted Answer

This problem is a classic example of how a seemingly impossible physics question can be cracked wide open with a simple shift in perspective. At first glance, it looks like a nightmare. We are given a velocity-time graph, but we have absolutely no mathematical equation for the curve. How on earth are we supposed to integrate an unknown function to find the distance?

The secret lies not in brute-force calculus, but in elegant geometry.

### Analyzing the Setup

Let's break down the journey of the two cars. Car A follows some unknown velocity profile, $v_A(t)$, starting from rest and eventually plateauing at $50	ext{ m/s}$.

Car B is the interesting one. It acts like a shadow of Car A, but with a glitch. 
1. **The Match:** From $0$ to $20	ext{ m/s}$, Car B perfectly matches Car A. They are side-by-side.
2. **The Pause:** The moment they hit $20	ext{ m/s}$, Car B stops accelerating. It cruises at a constant $20	ext{ m/s}$ for exactly $1	ext{ second}$.
3. **The Delay:** After that $1	ext{ second}$ nap, Car B wakes up and resumes accelerating exactly like Car A did. However, because it wasted a second, it is now permanently lagging behind Car A's velocity profile by exactly $1	ext{ second}$.

### The Geometric Translation

In kinematics, the area under a velocity-time ($v-t$) graph represents the total distance covered ($s = \int v \, dt$). Since we want to find the *difference* in distance, $\Delta s$, we need to find the difference in the areas under their respective curves.

Geometrically, this is simply the area enclosed *between* the blue curve of Car A and the red curve of Car B.

Before $20	ext{ m/s}$, the curves overlap perfectly. Area = $0$.
After $50	ext{ m/s}$, both cars are moving at a constant $50	ext{ m/s}$, so the curves overlap again. Area = $0$.

The only place where Car A gains ground is in the intermediate region, between $v = 20	ext{ m/s}$ and $v = 50	ext{ m/s}$.

### The Master Equation: A Paradigm Shift

Normally, to find the area between two curves, we use vertical strips and integrate with respect to time: $\int (v_A - v_B) \, dt$. But we don't know the functions!

Here is the master stroke: **Let's slice the area horizontally instead of vertically.**

If we integrate along the y-axis (the velocity axis), the area is given by the integral of the horizontal width of the region with respect to velocity:

$$ 	ext{Area} = \int_{v_1}^{v_2} (t_B - t_A) \, dv $$

Look at the horizontal width, $(t_B - t_A)$. This represents the time difference between Car B and Car A achieving the *same* velocity.

What did the problem tell us? After the pause, Car B follows Car A's profile with a delay of exactly $1	ext{ second}$. This means for *any* velocity between $20	ext{ m/s}$ and $50	ext{ m/s}$, Car B reaches that velocity exactly $1	ext{ second}$ after Car A.

The horizontal width is a constant!

$$ \Delta t = t_B(v) - t_A(v) = 1	ext{ s} $$

### Final Calculation

Our terrifying, impossible integral suddenly collapses into something a middle-schooler could solve. We just need to integrate the constant $1$ from the starting velocity of the gap ($20	ext{ m/s}$) to the ending velocity of the gap ($50	ext{ m/s}$).

$$ \Delta s = \int_{20}^{50} 1 \, dv $$

$$ \Delta s = [v]_{20}^{50} $$

$$ \Delta s = 50 - 20 = 30	ext{ m} $$

Car A covers exactly $30	ext{ meters}$ more than Car B. We didn't need to know if the curve was a parabola, an exponential, or a sine wave. The geometry of the constant time delay guaranteed the area would be exactly $30$. This is the true beauty of physics and mathematics working in perfect harmony.

Select Answer:

Visualized Solution

\text{The } v-t \text{ Graph and Distance}

\text{Velocity Profile of Car B}

\text{Area Between the Curves}

\text{Integration along } v\text{-axis}

\text{Constant Time Delay}

\text{Calculating } \Delta s

\text{Food for Thought}

The Sigma Insight: Motion Graphs

Analyzing the Setup

The Geometric Translation

The Master Equation

Final Calculation

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