Question

The equation of motion of a particle is s = 2t² + sin 2t, where s is in metres and t is in seconds. The velocity of the particle when its acceleration is 2 m/sec², is
(a) $\mathrm{\pi }+\sqrt{3}\mathrm{m}/\sec$

(b) $\frac{\mathrm{\pi }}{3}+\sqrt{3}\mathrm{m}/\sec$

(c) $\frac{2\mathrm{\pi }}{3}+\sqrt{3}\mathrm{m}/\sec$

(d) $\frac{\mathrm{\pi }}{3}+\frac{1}{\sqrt{3}}\mathrm{m}/\sec$

Answer 1

(b) $\frac{\mathrm{\pi }}{3}+\sqrt{3}\mathrm{m}/\sec$

$$\begin{gathered} \text{According to the question,} \\ s=2t^2+\sin 2t \\ \Rightarrow \frac{ds}{dt}=4t+2\cos 2t \\ \Rightarrow \frac{d^{2}s}{d{t}^2}=4-4\sin 2t \\ \Rightarrow 4-4\sin 2t=2 \\ \Rightarrow 4\sin 2t=2 \\ \Rightarrow \sin 2t=\frac{1}{2} \\ \Rightarrow 2t=\frac{\mathrm{\pi }}{6} \\ \mathrm{Now}, \\ \frac{ds}{dt}=4\left(\frac{\mathrm{\pi }}{12}\right)+2\cos \left(\frac{\mathrm{\pi }}{6}\right) \\ \Rightarrow \frac{ds}{dt}=\frac{\pi }{3}+\sqrt{3}\mathrm{m}/\sec \end{gathered}$$