Question

A car moves from X to Y with s uniform speed ${v_u}$ and returns to Y with a uniform speed ${v_d}$.The average speed for this round trip is
A. $\dfrac{{2{v_d}{v_u}}}{{{v_d} + {v_u}}}$
B. $\sqrt {{v_u}{v_d}} $
C. $\dfrac{{{v_d}{v_u}}}{{{v_d} + {v_u}}}$
D. $\dfrac{{{v_u} + {v_d}}}{2}$

Answer 1

Hint: Speed is how quick at a given moment something is going. Average speed measures the rate of the speed over the extent of a journey. Average speed is typically applied to vehicles. We can determine the average speed by dividing the total distance that a vehicle has travelled by the total amount of time it took to travel that distance.
Here we use the formula of distance, speed and time to determine the average speed.
Distance=speed × time.
Also, average speed = total distance/total time. The average speed in measured in $m/s$

Complete step by step answer:
Let the first half of distance $d$
 be covered in time ${t_1}$
 with speed ${v_u}$

We know that,
Distance=speed × time
Time= distance/speed
${t_1} = \dfrac{d}{{{v_u}}}$
Let the rest half of distance $d$ be covered in time ${t_2}$ with speed ${v_d}$
Again applying the formula of distance, speed and time we get-
${t_2} = \dfrac{d}
{{{v_d}}}$

Hence, total time is given by -
$
  t = {t_1} + {t_2} \\ = \dfrac{d}
{{{v_u}}} + \dfrac{d}
{{{v_d}}} \\ = \dfrac{{d{v_d} + d{v_u}}}
{{{v_u}{v_d}}} \\
 $

Total distance is the distance taken while going from one end to the other and also while returning.
Hence, total distance is given by -
$d = 2d$

Now, we can find the average speed.
Average speed =total distance/total time
Putting the values on the formula we get-
$ = \dfrac{{2d}}
{{\left( {\dfrac{{{v_d}d + {v_u}d}}
{{{v_u}{v_d}}}} \right)}} \\ = \dfrac{{2{v_u}{v_d}}}
{{({v_u} + {v_d})}} \\ $
Hence, the average speed during the complete journey is $\dfrac{{2{v_u}{v_d}}}{{({v_u} + {v_d})}}$

So, the correct answer is “Option A”.

Note:
Here the total distance should be taken as $2d only and not ${d_1} + {d_2}$otherwise we will not get the desired answer. Also the average speed is different from instantaneous speed. Instantaneous speed calculates speed of an object at a single moment in time. But mean speed is generally used in transportation.