Two projectiles are fired from ground with same initial speeds from same point at angles (45°+α)

Question

Two projectiles are fired from ground with same initial speeds from same point at angles\((45^\circ + \alpha)\) and \((45 ^\circ - \alpha)\) with horizontal direction. The ratio of their times of flights is

(1) 1

(2) \(\frac{1- tan\alpha}{1+ tan\alpha}\)

(3) \(\frac{1+ \sin2 \alpha}{1- \sin2\alpha}\)

(4) \(\frac{1+ \tan\alpha}{1-\tan \alpha}\)

Answer 1

Correct option is : (4) \(\frac{1+\tan \alpha}{1-\tan \alpha}\) 

\(\theta_1 = 45 +\alpha ; \ \theta _2 = 45 - \alpha\) 

Time of flight, \(T = \frac{2v\sin \theta}{g}\)

\(\frac{T_1}{T_2} = \frac{\sin(45 + \alpha)}{\sin(45- \alpha)}\)

\(\frac{T_1}{T_2} =\frac{\frac{1}{\sqrt2}\cos \alpha+\frac{1}{\sqrt2} \sin \alpha}{\frac{1}{\sqrt2}\cos \alpha - \frac{1}{\sqrt 2}\sin \alpha}\)

\(\frac{T_1}{T_2} = \frac{\cos \alpha + \sin \alpha }{cos \alpha - \sin \alpha} = \frac{1+ \tan \alpha}{1- \tan \alpha}\)