Acceleration Displacement Graph from Velocity Graph

❓ Question

The velocity-displacement graph describing the motion of a bicycle is shown in the figure.

Find the acceleration in different regions of motion.

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🖼 Question Image

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✍️ Short Explanation

This is a velocity-displacement graph problem.

👉 Use relation between velocity and displacement
👉 Acceleration is found using:

$$a=v\frac{dv}{dx}$$

instead of usual $v-u$ formulas.

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🔷 Step 1 — Important Formula 💯

For velocity-displacement graph:

$$a=v\frac{dv}{dx}$$

where:

$$\frac{dv}{dx}$$

is slope of $v$-$x$ graph.

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🔷 Step 2 — First Region $(0 \to 200\text{ m})$

From graph:

Velocity changes:

$$10 \to 50\text{ m/s}$$

Displacement:

$$0 \to 200\text{ m}$$

Slope:

$$\frac{dv}{dx} = \frac{50-10}{200}$$ $$=\frac{40}{200}$$ $$=0.2$$

Acceleration:

$$a=v(0.2)$$

Since velocity changes with position, acceleration is variable.

At any position in this region:

$$\boxed{a=0.2v}$$

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🔷 Step 3 — Second Region $(200 \to 400\text{ m})$

From graph:

Velocity is constant:

$$v=50\text{ m/s}$$

So:

$$\frac{dv}{dx}=0$$

Hence:

$$a=v\times0$$
$$\boxed{a=0}$$

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🔷 Step 4 — Direct Formula Check

Alternative method:

$$a=\frac{v^2-u^2}{2s}$$

For first region:

$$a=\frac{50^2-10^2}{2(200)}$$
$$=\frac{2500-100}{400}$$
$$=\frac{2400}{400}$$
$$=6\text{ m/s}^2$$

So acceleration in first section is constant:

$$\boxed{6\text{ m/s}^2}$$

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🔷 Step 5 — JEE Trap Alert 🚨

❌ Velocity-time graph samajh lena

❌ Slope ko directly acceleration maan lena

❌ $a=v\frac{dv}{dx}$ formula bhool jaana

Remember:

$$\text{Slope of }v-x\text{ graph} \neq \text{acceleration directly}$$

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✅ Final Answer

For:

$$0\to200\text{ m}$$
$$\boxed{a=6\text{ m/s}^2}$$

For:

$$200\to400\text{ m}$$
$$\boxed{a=0}$$

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