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A car moves from X to Y with a uniform speed $v_u$ and returns to X with a uniform speed $v_d$. The average speed for this round trip is
A
$\sqrt{v_u v_d}$
B
$\dfrac{v_d v_u}{v_d + v_u}$
C
$\dfrac{v_u + v_d}{2}$
D
$\dfrac{2 v_d v_u}{v_d + v_u}$
Detailed Solution
Average speed $= \dfrac{\text{total distance travelled}}{\text{total time taken}}$
Let $s$ be the distance from X to Y.
Time for the forward journey: $t_1 = \dfrac{s}{v_u}$
Time for the return journey: $t_2 = \dfrac{s}{v_d}$
Average speed $= \dfrac{s + s}{t_1 + t_2} = \dfrac{2s}{\dfrac{s}{v_u} + \dfrac{s}{v_d}}$
$= \dfrac{2s}{s\left(\dfrac{v_d + v_u}{v_u v_d}\right)}$
$= \dfrac{2 v_u v_d}{v_d + v_u}$
This is the harmonic mean of the two speeds, which always applies when equal distances are covered at two different speeds.
Let $s$ be the distance from X to Y.
Time for the forward journey: $t_1 = \dfrac{s}{v_u}$
Time for the return journey: $t_2 = \dfrac{s}{v_d}$
Average speed $= \dfrac{s + s}{t_1 + t_2} = \dfrac{2s}{\dfrac{s}{v_u} + \dfrac{s}{v_d}}$
$= \dfrac{2s}{s\left(\dfrac{v_d + v_u}{v_u v_d}\right)}$
$= \dfrac{2 v_u v_d}{v_d + v_u}$
This is the harmonic mean of the two speeds, which always applies when equal distances are covered at two different speeds.