Question

A body moves in a straight line with velocity \[{v_1}\] for \[\dfrac{1}{3}rd\] time and for remaining time with \[{v_2}\]. Find average velocity.
a. \[\dfrac{{{v_1}}}{3} + \dfrac{{2{v_2}}}{3}\]
b. \[\dfrac{{{v_1}}}{3} + \dfrac{{{v_2}}}{3}\]
c. \[\dfrac{{2{v_1}}}{3} + \dfrac{{{v_2}}}{3}\]
d. \[{v_1} + \dfrac{{2{v_2}}}{3}\]

Answer 1

Hint:The average velocity is calculated using the ratio of the net displacement to the total time of the journey. For a uniform motion in a straight line, the average velocity is equal to the average speed.

Formula used:
\[d = vt\]
Here d is the displacement, v is the velocity and t is the time of journey.
\[{v_{av}} = \dfrac{{{d_{net}}}}{{{t_{tot}}}}\]
here \[{v_{av}}\] is the average velocity.

Complete step by step solution:
Let the total time of journey is t. Then the time for which the body is moving with velocity \[{v_1}\] will be,
\[{t_1} = \dfrac{t}{3}\]
Using the displacement formula, the displacement during first interval of journey will be,
\[{d_1} = {v_1}{t_1}\]
\[\Rightarrow {d_1} = \dfrac{{{v_1}t}}{3}\]
Similarly, the time for which the body is moving with velocity \[{v_2}\]will be,
\[{t_2} = t - {t_1}\]
\[\Rightarrow {t_2} = t - \dfrac{t}{3}\]
\[\Rightarrow {t_2} = \dfrac{{3t - t}}{3}\]
\[\Rightarrow {t_2} = \dfrac{{2t}}{3}\]

Using the displacement formula, the displacement during second interval of journey will be,
\[{d_2} = {v_2}{t_2}\]
\[\Rightarrow {d_2} = \dfrac{{2{v_2}t}}{3}\]
So, the net displacement of the body during the motion is,
\[{d_{net}} = {d_1} + {d_2}\]
\[\Rightarrow {d_{net}} = \dfrac{{{v_1}t}}{3} + \dfrac{{2{v_2}t}}{3}\]
\[\Rightarrow {d_{net}} = \left( {\dfrac{{{v_1}}}{3} + \dfrac{{2{v_2}}}{3}} \right)t\]
Total time of journey is \[{t_{tot}} = t\]

Using the average velocity formula, the average velocity will be,
\[{v_{av}} = \dfrac{{{d_{net}}}}{{{t_{tot}}}}\]
\[\Rightarrow {v_{av}} = \dfrac{{\left( {\dfrac{{{v_1}}}{3} + \dfrac{{2{v_2}}}{3}} \right)t}}{t}\]
\[\therefore {v_{av}} = \left( {\dfrac{{{v_1}}}{3} + \dfrac{{2{v_2}}}{3}} \right)\]
Hence, the average velocity of the body during the complete journey is \[\dfrac{{{v_1}}}{3} + \dfrac{{2{v_2}}}{3}\].

Therefore, the correct option is A.

Note: If the motion is not uniform, then the average velocity will not be equal to the average speed. The value of the average velocity is equal or less than the average speed. The average velocity is equal when the motion is uniform in a straight line. Because the magnitude of the displacement is less than or equal to the distance covered, and the time of the journey is the same for both kinds of motion.