question_answer

A projectile is fired making an angle \[2\theta \]with horizontal with velocity 4 m/s. At any instant it makes an angle \[\theta ,\] then its velocity is:

A) \[4\cos \theta \]

B)  \[4(2cos\theta -sec\theta )\]

C)  \[(2cos\theta +4sec\theta )\]

D)  \[4(sec\theta +cos\theta )\]

Correct Answer: B

Solution :

 Key Idea: Horizontal component of velocity remains same. Let \[v\]be the velocity, when projected with angle \[\theta ,\] then equating the horizontal velocities in both the cases, we get               \[v=\cos \theta =u\cos 2\theta \] \[\Rightarrow \] \[v=\frac{u\cos 2\theta }{\cos \theta }\] where \[\sec \theta =\frac{1}{\cos \theta }\] \[\therefore \] \[v=u\cos 2\theta \sec \theta \] Using \[\cos 2\theta =2{{\cos }^{2}}\theta -1,\]we get Given,       \[u=4\,m/s,\,\]we get \[v=4(2co{{s}^{2}}\theta -1)sec\theta \] \[\Rightarrow \] \[v=4(2cos\theta -\sec \theta )\]