On a circular track, a motorcycle starts from point X and a truck starts from point Y , which is 500 m ahead of X in the direction of motion. The speed of the motorcycle is 20 m/s and that of the truck is 8 m/s. If the length of the track is 2.2 km, how much distance will the motorcycle have travelled when it overtakes the truck for the first time?

Correct Answer: The correct option is 833.33m.

Step-by-Step Explanation:

1. Understand the Given Information:

Speed of the motorcycle (v_m) = 20 m/s
Speed of the truck (v_t) = 8 m/s
Initial distance separation between the motorcycle and the truck = 500 m
Length of the circular track = 2.2 km = 2200 m

2. Determine the Relative Distance to Cover:

Since both the motorcycle and the truck are moving in the same direction, the motorcycle needs to cover the initial gap of 500 m between them to overtake the truck for the first time.

3. Calculate the Relative Speed:

When two objects move in the same direction, their relative speed is the difference between their speeds.

$Relative\; Speed=v_m-v_t$

$\text{Relative Speed}=20\text{m/s}-8\text{m/s}=12\text{m/s}$

4. Calculate the Time Taken to Overtake:

The time (t) taken by the motorcycle to overtake the truck is the initial separation distance divided by the relative speed:

$t=\frac{\text{Initial Distance Separation}}{\text{Relative Speed}}$

$t=\frac{500}{12}\text{seconds}=\frac{125}{3}\text{seconds}$

5. Calculate the Total Distance Travelled by the Motorcycle:

The total distance (d) travelled by the motorcycle during this time is:

$d=v_m\times t$

$d=20\times \frac{125}{3}=\frac{2500}{3}\text{m}\approx 833.33\text{m}$

Thus, the motorcycle will have travelled a distance of 833.33m when it overtakes the truck for the first time.