Question

A car travelling on a straight track moves with uniform velocity \(v_1\) for some time and with uniform velocity \(v_2\) for the next equal time. The average velocity of the car is

• A. \(\frac{v_1v_2}{2}\)
• B. \(\frac{v_1v_2}{4}\)
• C. \(\frac{v_1 + v_2}{2}\)
• D. \(\frac{v_1 - v_2}{2}\)

Hint:

For equal time intervals, average velocity is arithmetic mean of velocities.

Answer 1

The Correct Option is C

To solve this problem, we need to understand the concept of average velocity when an object travels with different velocities over equal time intervals.

• The car travels with velocity \(v_1\) for a certain amount of time, say \(t\).
• It then travels with velocity \(v_2\) for the same time interval \(t\).

Average velocity is calculated as the total displacement divided by the total time taken. Let's derive this step-by-step:

1. Calculate the displacement during the first part of the journey:
   • Displacement \(d_1 = v_1 \cdot t\)
2. Calculate the displacement during the second part of the journey:
   • Displacement \(d_2 = v_2 \cdot t\)
3. Total displacement for the entire journey:
   • \(d_{\text{total}} = d_1 + d_2 = v_1 \cdot t + v_2 \cdot t = (v_1 + v_2) \cdot t\)
4. Total time taken for the journey:
   • \(t_{\text{total}} = t + t = 2t\)
5. Average velocity \(v_{\text{avg}}\) is given by:
   • \(v_{\text{avg}} = \frac{d_{\text{total}}}{t_{\text{total}}} = \frac{(v_1 + v_2) \cdot t}{2t}\)
   • On simplifying, we get \(v_{\text{avg}} = \frac{v_1 + v_2}{2}\)

Thus, the correct answer, showing that the average velocity is the arithmetic mean of the two velocities, is \(\frac{v_1 + v_2}{2}\).

Other options can be ruled out because:

• \(\frac{v_1v_2}{2}\) and \(\frac{v_1v_2}{4}\) involve multiplication rather than addition, which is incorrect for calculating average velocity.
• \(\frac{v_1 - v_2}{2}\) involves subtraction, which also does not apply to this scenario of averaging two velocities.

 

Therefore, the correct and final answer is \(\frac{v_1 + v_2}{2}\).