JEE Main 2024 — Circular Motion Question with Solution

To determine the distance traveled by the tips of the second hand and the minute hand, we need to calculate the circumference of the circles they make. Let's start by calculating the distances traveled by both hands over a period of 30 minutes.

First, the circumference formula is given by:

$$C=2\pi r$$

For the second hand:

The length of the second hand is $$75\text{cm}$$. The second hand completes one full revolution every 60 seconds, so in 1 minute, the second hand travels:

$$\text{Distance per minute}=2\pi \times 75\text{ cm}$$

In 30 minutes, the second hand will travel:

$$\text{Total distance traveled by second hand}=30\times 2\pi \times 75\text{ cm}$$

Now, let's calculate it with $$\pi =3.14$$:

$$\text{Total distance traveled by second hand}=30\times 2\times 3.14\times 75\text{ cm}$$

$$\text{Total distance traveled by second hand}=30\times 471\text{ cm}=14130\text{ cm}$$

For the minute hand:

The length of the minute hand is $$60\text{ cm}$$. The minute hand completes one full revolution every 60 minutes, so in 30 minutes, the minute hand travels:

$$\text{Total distance traveled by minute hand}=0.5\times 2\pi \times 60\text{ cm}$$

Again, let's calculate it with $$\pi =3.14$$:

$$\text{Total distance traveled by minute hand}=0.5\times 2\times 3.14\times 60\text{ cm}$$

$$\text{Total distance traveled by minute hand}=0.5\times 376.8\text{ cm}=188.4\text{ cm}$$

The difference in distance $$x$$ is:

$$x=14130\text{ cm}-1884\text{ cm}$$

$$x=13941.6\text{ cm}$$

Convert this distance into meters:

$$x=13941.6\text{ cm}\times \frac{1\text{ meter}}{100\text{ cm}}$$

$$x\approx 139.4\text{ meters}$$

Thus, the value of $$x$$ in meters is nearly 139.4 meters, making the correct option:

Option C