Question

A particle travels first half of the total distance with speed $v_1$. In second half distance, constant speed in $\frac{1}{3}rd$ time is $v_2$ and in remaining $\frac{2}{3}rd$ time constant sped is $v_3$. Find average speed during the complete journey.

Answer 1

Average speed $\left(v_{avg}\right)=\frac{Distanc{e}_{\left(Total\right)}}{Tim{e}_{\left(Total\right)}}=\frac{\Sigma v_i{t}_i}{\Sigma t_i}$

So, Average speed $=\frac{v_1\left(\frac{t}{2}\right)+v_2\left(\frac{t}{2}\right)\left(\frac{1}{3}\right)+v_3\left(\frac{t}{2}\right)\left(\frac{2}{3}\right)}{t}$

$\Rightarrow v_{avg}=$ $\frac{v_1}{2}$ $+\frac{v_2}{6}$ $\frac{v_3}{3}$ $=\frac{(3v_1+v_2+2v_3)}{6}$