Question

A car travelling on a straight track moves with uniform velocity of ${v}_{1}$ for some time and then with uniform velocity ${v}_{2}$ for the next equal time. The average velocity of the car is?
A. $\dfrac{{v}_{1}{v}_{2}}{2}$
B. $\dfrac{{v}_{1}{v}_{2}}{4}$
C. $\dfrac{{v}_{1}+{v}_{2}}{2}$
D. $\dfrac{{v}_{1}-{v}_{2}}{2}$

Answer 1

Hint: Our first step to solve this question should be to find the distance travelled by the car in each of the intervals. Then, as the time intervals are equal, we can find the average velocity by taking the ratio of total distance travelled by the car to the total time taken.

Complete step by step solution:
We have been given that the car travelled for equal time intervals with velocities ${v}_{1}$ and ${v}_{2}$.
Let the time interval for which the car travelled be $t$.
The by using the formula, $distance=speed\times time$, distance travelled by the car in first interval with velocity ${v}_{1}$ will be ${d}_{1}={v}_{1}t$ ………. (i).
Similarly, distance travelled by the car in the second equal interval of time with the velocity ${v}_{2}$ will be ${d}_{2}={v}_{2}t$ ……….(ii)
Therefore, total distance travelled by the car will be ${d}_{1}+{d}_{2}$ and total time of travel will be 2t.
Then using equation (i) and (ii), average velocity of the car, $v=\dfrac{\text{total distance travelled}}{\text{total time taken}} =\dfrac{{v}_{1}t+{v}_{2}t}{2t}=\dfrac{t({v}_{1}+{v}_{2})}{2t}=\dfrac{{v}_{1}+{v}_{2}}{2}$
Hence, option c is the correct answer.

Note: The time intervals have been given equal, and also the direction of motion didn’t change over time. So, the net displacement will be the same as the total distance travelled. In some questions, the time intervals and the direction of motion may not be the same and hence the net displacement may change.