Question

A car moves from $X$ to $Y$ with a uniform speed $v_{u}$ and returns to $Y$ with a uniform speed $v_{d}$. The average speed for this round trip is

• A. $\frac{2 v_{d} v_{u}}{v_{d}+v_{u}}$
• B. $\sqrt{\nu_{u} v_{d}}$
• C. $\frac{v_{d} v_{u}}{v_{d}+v_{u}}$
• D. $\frac{v_{u}+v_{d}}{2}$

Answer 1

Answer: (A)

$\frac{2 v_{d} v_{u}}{v_{d}+v_{u}}$

Answer 2

Let $t_{1}$ and $t_{2}$ be times taken by the car to go from $X$ to $Y$ and then from $Y$ to $X$ respectively. Then, $t_{1}+t_{2}=\frac{X Y}{v_{u}}+\frac{X Y}{v_{d}}=X Y \cdot\left(\frac{v_{u}+v_{d}}{v_{u} v_{d}}\right)$
Total distance travelled
$=X Y+X Y=2 X Y$
Therefore, average speed of the car for this round trip is
$v_{ av } =\frac{2 X Y}{X Y\left(\frac{v_{u}+v_{d}}{v_{u} v_{d}}\right)}$
or $v_{ av }=\frac{2 v_{u} v_{d}}{v_{u}+v_{d}}$