Question

A body travelling along a straight line traversed one third of the total distance with a velocity of 4 m/s.The remaining part of the distance was covered with a velocity 2 m/s for half the time and with velocity 6 m/s for the other half of time.The mean velocity averaged over the whole time of motion is
(a) 5 m/s
(b) 4 m/s
(c) 4.5 m/s
(d) 3.5 m/s

Answer 1

$$\begin{gathered} \mathrm{Let} \\ \mathrm{Total}\mathrm{distance}=\mathrm{s} \\ \mathrm{Time}\mathrm{taken}\mathrm{for}\mathrm{first}\mathrm{half}\mathrm{distance}: \\ \mathrm{Distance},{\mathrm{s}}_1=\frac{\mathrm{s}}{2} \\ {\mathrm{t}}_1=\frac{\mathrm{s}/2}{4} \\ =\frac{\mathrm{s}}{8} \\ \mathrm{Let} \\ \mathrm{Time}\mathrm{taken}\mathrm{for}\mathrm{remaining}\mathrm{distance}={\mathrm{t}}_2+{\mathrm{t}}_3 \\ \mathrm{Distance}\mathrm{travelled}\mathrm{with}2\mathrm{m}/\mathrm{s},{\mathrm{s}}_2=2{\mathrm{t}}_2 \\ \mathrm{Distance}\mathrm{travelled}\mathrm{with}6\mathrm{m}/\mathrm{s},{\mathrm{s}}_3=6{\mathrm{t}}_3 \\ \mathrm{Remaining}\mathrm{distance}: \\ \frac{\mathrm{s}}{2}={\mathrm{s}}_2+{\mathrm{s}}_3 \\ \frac{\mathrm{s}}{2}=2{\mathrm{t}}_2+6{\mathrm{t}}_3 \\ \mathrm{But},{\mathrm{t}}_2={\mathrm{t}}_3 \\ \mathrm{Hence}, \\ \frac{\mathrm{s}}{2}=2{\mathrm{t}}_2+6{\mathrm{t}}_2 \\ \frac{\mathrm{s}}{2}=8{\mathrm{t}}_2 \\ {\mathrm{t}}_2=\frac{\mathrm{s}}{16} \\ {\mathrm{t}}_3={\mathrm{t}}_2=\frac{\mathrm{s}}{16} \\ \mathrm{Average}\mathrm{speed}\mathrm{for}\mathrm{the}\mathrm{whole}\mathrm{motion}: \\ \mathrm{v}=\frac{{\mathrm{s}}_1+{\mathrm{s}}_2+{\mathrm{s}}_3}{{\mathrm{t}}_1+{\mathrm{t}}_2+{\mathrm{t}}_3} \\ =\frac{\mathrm{s}}{{\displaystyle \frac{\mathrm{s}}{8}}+{\displaystyle \frac{\mathrm{s}}{16}}+{\displaystyle \frac{\mathrm{s}}{16}}} \\ =\frac{\mathrm{s}}{{\displaystyle \frac{\mathrm{s}}{8}+\frac{\mathrm{s}}{8}}} \\ =\frac{\mathrm{s}}{{\displaystyle \frac{\mathrm{s}}{4}}} \\ \mathbf{=}\mathbf{4}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s} \\ \mathbf{Ans}\mathbf{:}\mathbf{}\mathbf{Option}\mathbf{}\mathbf{-}\mathbf{}\mathbf{\left(}\mathbf{b}\mathbf{\right)} \end{gathered}$$