JEE Physics 2023 Question with Solution

Given: The rider covers the first half of the total distance with speed $5 \, \text{m/s}$. In the remaining half, he moves at $10 \, \text{m/s}$ for half of that interval of time and at $15 \, \text{m/s}$ for the other half.

Find: The value of $x$ if the average speed over the whole journey is $\frac{x}{7} \, \text{m/s}$.

Let $AB = x$ and $BC = x$. So the total distance is

For the first half,

For motion along $BC$, let the total time be $t$. Since the rider moves for equal time intervals at $10 \, \text{m/s}$ and $15 \, \text{m/s}$, the distances covered are

Hence,

So,

Therefore, total time of motion is

Average speed over the whole journey is

Now simplify:

Thus,

Given that average speed is $\frac{x}{7} \, \text{m/s}$, we get

Hence,

Therefore, the required value is $50$.

Given: The second half of the distance is covered using speeds $10 \, \text{m/s}$ and $15 \, \text{m/s}$ for equal times.

Find: The overall average speed.

For equal time intervals, the average speed on that part is the arithmetic mean:

So the rider covers the second half distance $x$ with effective speed $\frac{25}{2} \, \text{m/s}$.

Thus the times for the two halves are

Total distance is $2x$, so average speed is

Comparing with $\frac{x}{7} \, \text{m/s}$ gives

Therefore, the correct answer is $50$.