Question

a student is at distance of 50 meters from the bus as soon as the bus starts its motion with an acceleration of 1m/s the students starts running towards the bus with a uniform velocity u . assuming the motion to be a straight road the minimum value f u , so that the students is able to catch the bus is ?

Answer 1

$$\begin{gathered} Let \\ \mathrm{t}=Timeintervaltomeetstudentandbus \\ Conditiontomeet: \\ Distancetravelledbystudent=Distancetravelledbybus+50\mathrm{m} \\ ut=\frac{1}{2}a{t}^2+50 \\ \mathrm{u}=\frac{1}{2}at+\frac{50}{\mathrm{t}} \\ forminimumvalueof\mathrm{u}, \\ \frac{du}{dt}=0 \\ \frac{du}{dt}=\frac{1}{2}\mathrm{a}-\frac{50}{{\mathrm{t}}^2} \\ 0=\frac{1}{2}\mathrm{a}-\frac{50}{{\mathrm{t}}^2} \\ \frac{50}{{\mathrm{t}}^2}=\frac{\mathrm{a}}{2} \\ {\mathrm{t}}^2=\frac{100}{\mathrm{a}} \\ =\frac{100}{1} \\ \mathrm{t}=10\mathrm{s} \\ Hence, \\ \mathrm{u}=\frac{1}{2}at+\frac{50}{\mathrm{t}} \\ =\left[\frac{1}{2}\times 1\times 10\right]+\frac{50}{10} \\ =10\mathrm{m}/\mathrm{s} \\ \mathbf{Ans}\mathbf{:}\mathbf{}\mathbf{10}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s} \end{gathered}$$