Question

On a circular track, two cyclists, Abhijit and Vani, start moving in opposite directions from a point. Abhijit moves with a constant speed. Vani starts with a constant acceleration from rest. They meet again on the track with the same speed. Which of the following is correct?

• A. Abhijit travelled double the distance travelled by Vani.
• B. Abhijit travelled half the distance travelled by Vani.
• C. Abhijit travelled the same distance travelled by Vani.
• D. Abhijit travelled 4/3 of the distance travelled by Vani.

Hint:

For any motion starting from rest with constant acceleration, the average velocity is exactly half of the final velocity.
Since Abhijit travels at a constant velocity equal to Vani's final velocity, Abhijit's average speed is twice that of Vani's.
Consequently, for the same time interval, Abhijit must travel twice the distance.

Answer 1

The Correct Option is A

To determine how the distances traveled by Abhijit and Vani relate to each other when they meet again on a circular track, we must analyze their motions:

1. Understand the Motion:
   • Abhijit travels with a constant speed \(v_a\). Since he maintains a constant speed, the distance he travels is directly proportional to the time.
   • Vani starts from rest with a constant acceleration \(a\). Her speed therefore increases linearly with time, meaning at any time \(t\), her speed \(v_v\) is given by \(v_v = a \cdot t\).
2. Condition when they meet again:
   • They meet on the track when they have covered a whole number multiple of the circular track's circumference combined. Since they start from the same point and in opposite directions, we infer this condition: Abhijit’s distance + Vani’s distance is a multiple of the track's circumference (\(C\)).
   • They have the same speed when they meet. Let's denote this speed as \(v\). Therefore, when they meet, \(v = v_a = a \cdot t\).
3. Calculate the distances:
   • The distance Abhijit covers: \(d_a = v_a \cdot t\)
   • The distance Vani covers from rest with acceleration: \(d_v = \frac{1}{2} \cdot a \cdot t^2\)
   • Given that at time \(t\), \(v_a = a \cdot t\), we substitute \(t = \frac{v_a}{a}\) into Vani's distance formula:
   • \(d_v = \frac{1}{2} \cdot a \cdot \left(\frac{v_a}{a}\right)^2 = \frac{v_a^2}{2a}\)
   • We know when they meet, both have speed \(v\) which is also \(v_a\). Thus, equating Vani's speed equation to Abhijit's speed \(v_a = a \cdot \frac{v_a}{a}\), thus:
   • \(d_a = v_a \cdot t = v_a \cdot \frac{v_a}{a} = \frac{v_a^2}{a}\)
4. Compare distances:
   • Comparing \(d_a\) and \(d_v\), we have:
   • \(d_a = \frac{v_a^2}{a},\) and \(d_v = \frac{v_a^2}{2a}\)
   • This shows:
   • \(d_a = 2\cdot d_v\)

Thus, the correct statement is: "Abhijit travelled double the distance travelled by Vani."