Two projectiles are fired from ground with same initial speeds from same point at angles (45° + ) and (45° – ) with horizontal direction. The ratio of their times of flights is [2025]

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Q.
Two projectiles are fired from ground with same initial speeds from same point at angles (45° + $α$ ) and (45° – $α$ ) with horizontal direction. The ratio of their times of flights is [2025]

1 1

2 $\frac{1 - \tan α}{1 + \tan α}$

3 $\frac{1 + \sin 2 α}{1 - \sin 2 α}$

4 $\frac{1 + \tan α}{1 - \tan α}$

Ans.
(4)

Time of flight for $1^{st}$ projectile,

$T_{1} = \frac{2 u \sin (45 + α)}{g}$

Time of flight for $2^{nd}$ projectile,

$T_{2} = \frac{2 u \sin (45 - α)}{g}$

$\frac{T_{1}}{T_{2}} = \frac{\sin (45 + α)}{\sin (45 - α)}$

$\frac{T_{1}}{T_{2}} = \frac{\frac{1}{\sqrt{2}} \cos α + \frac{1}{\sqrt{2}} \sin α}{\frac{1}{\sqrt{2}} \cos α - \frac{1}{\sqrt{2}} \sin α}$

$\frac{T_{1}}{T_{2}} = \frac{\cos α + \sin α}{\cos α - \sin α} = \frac{1 + \tan α}{1 - \tan α}$

Solution

Q.
Two projectiles are fired from ground with same initial speeds from same point at angles (45° + $α$ ) and (45° – $α$ ) with horizontal direction. The ratio of their times of flights is [2025]

1 1

2 $\frac{1 - \tan α}{1 + \tan α}$

3 $\frac{1 + \sin 2 α}{1 - \sin 2 α}$

4 $\frac{1 + \tan α}{1 - \tan α}$

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Solution

Q. Two projectiles are fired from ground with same initial speeds from same point at angles (45° + α) and (45° – α) with horizontal direction. The ratio of their times of flights is [2025]

1 1

2 1–tanα1+tanα

3 1+sin2α1–sin2α

4 1+tanα1–tanα

Ans. (4) Time of flight for 1st projectile, T1=2u sin(45+α)g Time of flight for 2nd projectile, T2=2u sin(45–α)g T1T2=sin(45+α)sin(45–α) T1T2=12cosα+12sinα12cosα–12sinα T1T2=cosα+sinαcosα–sinα=1+tanα1–tanα

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Q.
Two projectiles are fired from ground with same initial speeds from same point at angles (45° + $α$ ) and (45° – $α$ ) with horizontal direction. The ratio of their times of flights is [2025]

2 $\frac{1 - \tan α}{1 + \tan α}$

3 $\frac{1 + \sin 2 α}{1 - \sin 2 α}$

4 $\frac{1 + \tan α}{1 - \tan α}$