Question

A particle moves in a straight line it moves for half of total time with velocity V1 and another half time it moves half of distance with velocity V 2 and half of distance with velocity v 3 . find average velocity during entire Motion ?

Answer 1

$\langle|\vec{v}|\rangle=\dfrac{\displaystyle\sum_{i=1}^{n}d_i}{\displaystyle\sum_{i=1}^{n}t_i}\\=\dfrac{d_1+d_{2,3}}{t}\\=\dfrac{v_1\cdot\dfrac{t}{2}+{\langle|\vec{v}|\rangle}_{2,3}\dfrac{t}{2}}{t}\\=\dfrac{v_1}{2}+\dfrac{1}{2}\dfrac{\displaystyle\sum_{i=1}^{m}d_i}{\displaystyle\sum_{i=1}^{m}t_i}\\=\dfrac{v_1}{2}+\dfrac{1}{2}\dfrac{d_{2,3}}{t_2+t_3}\\=\dfrac{v_1}{2}+\dfrac{1}{2}\dfrac{d_{2,3}}{\dfrac{d_{2,3}}{2v_2}+\dfrac{d_{2,3}}{2v_3}}\\=\dfrac{v_1}{2}+\dfrac{1}{\dfrac{v_2+v_3}{v_2v_3}}\\=\dfrac{v_1}{2}+\dfrac{v_2v_3}{v_2+v_3}$