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

Options

• \[\pi + \sqrt{3} m\ /\sec\]

• \[\frac{\pi}{3} + \sqrt{3} m/\sec\]

• \[\frac{2\pi}{3} + \sqrt{3} m/\sec\]

• \[\frac{\pi}{3} + \frac{1}{\sqrt{3}} m/\sec\]

Answer 1

 \[\frac{\pi}{3} + \sqrt{3} \ m/\sec\]

\[\text { According to the question },\] 

\[s = 2 t^2  + \sin  2t\] 

\[ \Rightarrow \frac{ds}{dt} = 4t + 2  \cos  2t\] 

\[ \Rightarrow \frac{d^2 s}{d t^2} = 4 - 4  \sin  2t\] 

\[ \Rightarrow 4 - 4  \sin  2t = 2\] 

\[ \Rightarrow 4  \sin  2t = 2\] 

\[ \Rightarrow \sin  2t = \frac{1}{2}\] 

\[ \Rightarrow 2t = \frac{\pi}{6}\] 

\[\text { Now,} \] 

\[\frac{ds}{dt} = 4\left( \frac{\pi}{12} \right) + 2  \cos\left( \frac{\pi}{6} \right)\] 

 

\[ \Rightarrow \frac{ds}{dt} = \frac{\pi}{3} + \sqrt{3}  m/\sec\]