Two Stones Falling Problem Height of Building

❓ Question

A stone is dropped from the top of a building. When it crosses a point 5 m below the top, another stone starts to fall from a point 25 m below the top. Both stones reach the bottom of the building simultaneously. Find the height of the building.

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🖼 Question Image

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✍️ Short Explanation

This is a relative motion + free fall problem.

👉 Both stones move under gravity
👉 Final arrival time is same
👉 Use equations of motion separately for both stones.

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🔷 Step 1 — Motion of First Stone

First stone starts from top.

When it falls:

$$5\text{ m}$$

its velocity becomes:

Using:

$$v^2=u^2+2gs$$
$$v^2=0+2g(5)$$
$$v^2=10g$$
$$v=\sqrt{10g}$$

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🔷 Step 2 — Second Stone Starts

At this instant:

👉 First stone is 5 m below top
👉 Second stone starts from 25 m below top

Distance between stones:

$$25-5=20\text{ m}$$

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🔷 Step 3 — Let Height of Building be $H$

Remaining distance for first stone:

$$H-5$$

Remaining distance for second stone:

$$H-25$$

Both reach simultaneously.

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🔷 Step 4 — Apply Equations of Motion

For first stone:

Initial velocity:

$$u=\sqrt{10g}$$

Distance:

$$H-5$$

So:

$$H-5=\sqrt{10g}t+\frac{1}{2}g{t}^2$$

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For second stone:

Initial velocity:

$$u=0$$

Distance:

$$H-25$$

So:

$$H-25=\frac{1}{2}g{t}^2$$

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🔷 Step 5 — Subtract Equations

Subtract second equation from first:

$$(H-5)-(H-25)=\sqrt{10g}t$$
$$20=\sqrt{10g}t$$
$$t=\frac{20}{\sqrt{10g}}$$

Now use:

$$H-25=\frac{1}{2}g{t}^2$$
$$H-25=\frac{1}{2}g{\left(\frac{20}{\sqrt{10g}}\right)}^2$$
$$H-25=\frac{1}{2}g\cdot \frac{400}{10g}$$
$$H-25=20$$
$$H=45\text{ m}$$

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🔷 Step 6 — JEE Trap Alert 🚨

❌ Both stones ka starting time same maan lena

❌ First stone ki initial velocity ignore kar dena

❌ Relative distance galat lena

Remember:

$$\text{First stone already has velocity when second starts}$$

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✅ Final Answer

$$\boxed{{\displaystyle 45\text{ m}}}$$

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📚 Related Topics

• Relative Motion with Upward Accelerating Frame
• Momentum Height Graph for Vertical Motion
• Acceleration Velocity Displacement vs Time Graphs