Question

A car travels from place A to place B at $20$km/hour and returns at $30$km/hour. The average speed of the car the whole journey is

Answer 1

Hint: Concept of average speed is to be used. Average speed is the ratio of total distance covered to total time taken.
Average speed, $v_{av}={\displaystyle \frac{s_1+s_2}{t_1+t_2}}$
Where in time $t_1$, speed is $s_1$
For time $t_2$, speed is $s_2$

Complete step by step answer:
When the car travels from place A to B,
speed $=20$km/hour
Let the distance between place A and B be s km.
So, for journey from place A to place B,
distance, $s_1=skm$
speed, $v_1=20$km/hour
time taken, $t_1={\displaystyle \frac{dis\tan ce}{speed}}$
$$\begin{gathered} \Rightarrow timetaken,t_1={\displaystyle \frac{s_1}{v_1}} \\ \Rightarrow t_1={\displaystyle \frac{s}{20}}hours....(1) \end{gathered}$$
Now, for the journey from place B to place A i.e. during return distance will be the same.
So, distance, $s_2=skm$
speed, $v_2=30$ km/hr
time taken, $t_2={\displaystyle \frac{s_2}{v_2}}\Rightarrow t_2={\displaystyle \frac{s}{30}}$ hour
Now, average speed = $$\begin{gathered} ={\displaystyle \frac{Totaldis\tan ce}{Totaltime}} \\ i.e.v_{au}={\displaystyle \frac{s_1+s_2}{t_1+t_2}} \end{gathered}$$
Putting the values, we get
$v_{av}={\displaystyle \frac{s+s}{{\displaystyle \frac{s}{20}}+{\displaystyle \frac{s}{30}}}}$
$\Rightarrow v_{av}={\displaystyle \frac{2s}{{\displaystyle \frac{30s+20s}{20\times 30}}}}$
$$\begin{gathered} \Rightarrow v_{av}=2s\times {\displaystyle \frac{600}{50s}} \\ \Rightarrow v_{av}=2\times 12 \\ \Rightarrow v_{av}=24km/hour \end{gathered}$$
So, the average speed of the car through the whole journey is $24km/hour.$

Note:
Also when the distances are same, the average speed is given by
$v_{av}={\displaystyle \frac{2v_1{v}_2}{v_1+v_2}}$
as $v_1=20km/hour{v}_2=30km/hour$
So, $$\begin{gathered} v_{av}={\displaystyle \frac{2\times 20\times 30}{20+30}} \\ ={\displaystyle \frac{1200}{50}} \\ =24km/hr \end{gathered}$$