Question

A particle covers half of its distance with speed ${v_1}$ and the rest half distance with speed ${v_2}$. Its average speed during the complete journey is:
A) $\dfrac{{{v_1} + {v_2}}}{2}$
B) $\dfrac{{{v_1}{v_2}}}{{{v_1} + {v_2}}}$
C) $\dfrac{{2{v_1}{v_2}}}{{{v_1} + {v_2}}}$
D) \[\dfrac{{{v_1}^2{v_2}^2}}{{{v_1}^2 + {v_2}^2}}\]

Answer 1

Hint: In this question we have to find the average speed of the particle. For this we are going to use the formula of average speed. Before finding the average speed we will find speed for first and second half both and then we will plug them to find the average speed.

Complete step by step solution:
The first half of the distance is covered with speed ${v_1}$ and the rest half distance with speed ${v_2}$. Let the average speed is v. If the time taken to cover the first half distance is ${t_1}$ and the time taken to cover the second half distance is ${t_2}$. The formula of speed is as follows,
$speed = \dfrac{{distance}}{{time}}$
Using the formula of speed for first half of the distance ${v_1} = \dfrac{x}{{{t_1}}}$
Finding the value of ${t_1}$ from above equation,
$\Rightarrow {t_1} = \dfrac{x}{{{v_1}}}$
Speed of the second half of the distance ${v_2} = \dfrac{x}{{{t_2}}}$
Finding the value of ${t_2}$from above equation,
$\Rightarrow {t_2} = \dfrac{x}{{{v_2}}}$
Total time \[t = {t_1} + {t_2}\]
Total distance =2x
The average speed of the particle is given by following formula,
$\Rightarrow {\text{average speed = }}\dfrac{{{\text{total distance}}}}{{{\text{total time}}}}$
Putting the values of total distance and total time in above equation,
$\Rightarrow {\text{average speed = }}\dfrac{{{\text{2x}}}}{{{{\text{t}}_1}{\text{ + }}{{\text{t}}_2}}}$
Putting the values of
$\Rightarrow {\text{average speed v = }}\dfrac{{{\text{2x}}}}{{\dfrac{x}{{{v_1}}}{\text{ + }}\dfrac{x}{{{v_2}}}}}$
$\Rightarrow {\text{average speed v = }}\dfrac{{{\text{2x}}}}{{x\left( {\dfrac{1}{{{v_1}}}{\text{ + }}\dfrac{1}{{{v_2}}}} \right)}}$
$\Rightarrow {\text{average speed v = }}\dfrac{{\text{2}}}{{\left( {\dfrac{1}{{{v_1}}}{\text{ + }}\dfrac{1}{{{v_2}}}} \right)}}$
\[\Rightarrow {\text{average speed v = }}\dfrac{{\text{2}}}{{\left( {\dfrac{{{v_2} + {v_1}}}{{{v_1}{v_2}}}} \right)}}\]
\[\Rightarrow {\text{average speed v = }}\dfrac{{{\text{2}}{v_1}{v_2}}}{{{v_2} + {v_1}}}\]
Result- Hence, from above calculation the average speed of the particle is, \[{\text{v = }}\dfrac{{{\text{2}}{v_1}{v_2}}}{{{v_2} + {v_1}}}\].

Note: In such types of questions there might be a case where we will have to find acceleration also. In that case we will have to differentiate the average speed with respect to time. That case will become quite tedious because it is also a function of time. This case where we have to find the average speed only is a bit easier than that. But for this case also we should know the formula of speed and average speed.