Question

An airplane flies from a town $A$ to town $B$ when there is no wind and takes a total time $T_o$ for a return trip. When there is a wind blowing with speed less than that of airplane in a direction from town $A$ to town $B$, the total time for a similar return trip is $T_w$. Which of the following would satisfy this condition?

• A. $T_o<T_w$
• B. $T_o>T_w$
• C. $T_o=T_w$
• D. The result depends on the wind velocity between the towns

Answer 1

The correct option is A $T_o<T_w$

Let the distance between town $A$ and $B$ be $S$
Let $v$ and $v_w$ be the velocities of airplane and wind respectively.
Case 1: When there is no wind
Time taken by the airplane for the round trip $=\frac{2s}{v}=T_o$ (where $v$ is velocity of airplane)
Case 2: When wind is blowing from $A$ to $B$
Time taken by the airplane for the round trip, $T_w=\frac{s}{(v+v_w)}+\frac{s}{(v-v_w)}=s\left[\frac{2v}{(v^2-v_w^2)}\right]=\frac{2s}{v\left[\frac{(v^2-v_w^2)}{v^2}\right]}$ (where $v_w$ is the velocity of wind)
$T_w=T_o\left[\frac{1}{(1-(v_w/v{)}^2)}\right]\text{Since}{v}_w<v\Rightarrow T_w>T_o$