Question

A body travels the first half of the total distance with velocity $v_1$ and the second half with velocity $v_2$. Calculate the average velocity.

• A. $\frac{v_1+v_2}{2}$
• B. $\sqrt{v_1{v}_2}$
• C. $\sqrt{2v_1{v}_2}$
• D. $\frac{2v_1{v}_2}{v_1+v_2}$

Answer 1

The correct option is D $\frac{2v_1{v}_2}{v_1+v_2}$


From above diagram,

total distance $=2d$

Body travels $d$ distance from $\text{A}$ to $\text{C}$ in $t_1$ time with velocity $v_1$.

$\therefore t_1=\frac{d}{v_1}$

Similarly, body travels $d$ distance from $\text{C}$ to $\text{B}$ in $t_2$ time with velocity $v_2$.

$\therefore t_2=\frac{d}{v_2}$

So, Average velocity $(<v>)$ is defined as,

$<v>=\frac{\text{total displacement}}{\text{total time}}=\frac{2d}{t_1+t_2}$

$\Rightarrow <v>=\frac{2d}{\frac{d}{v_1}+\frac{d}{v_2}}=\frac{2v_1{v}_2}{v_1+v_2}$

Hence, option $\left(d\right)$ is correct.