Distance between two stations $A$ and $B$ is $778$km. A train covers the journey from $A$ to $B$ at a uniform speed of $84$km per hour and returns back to $A$ with a uniform speed of $56$km per hour. Find the average speed of the train during the whole journey?
The correct answer is:
Let distance between $A$ and $B$ be $x$
Time taken for travelling from $A$ to $B$ is $\frac{x}{84}$
Time taken for travelling from $B$ to $A$ is $\frac{x}{56}$
Total distance travelled is $x+x=2x$
Total time taken is $\frac{x}{84}+\frac{x}{56}$
Average speed is $67.2$
I know speed is distace/time and
I dont know if it may sound stupid but
what i thought of was
$$avg=\frac{\text{speed}_1+\text{speed}_2}{2} \tag{This is how we calculate average}$$
$$\frac{84+56}{2}=70$$
why is it giving wrong answer?
Also do I need to revise my physics concepts or maths or both?
Best Answer
An average speed is a ratio of a distance travelled to a travelling time.
An average distance in the first pass is $v_1 = s/t_1$ and an average speed in the returning pass on the same distance is $v_2 = s/t_2$.
The average speed on the whole travel is a total distance to total time used: $$v=\frac{s+s}{t_1+t_2}$$ which results in: $$v = \frac 2{\frac{t_1+t_2}s} =\frac 2{\frac {t_1} s+\frac {t_2} s}=\frac 2{\frac 1{v_1}+\frac 1{v_2}}$$