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

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Correct answer: A. (a) To find the ratio of the times of flight for two projectiles fired from the ground with the same initial speed but at different angles, we follow these steps: Projectile Angles: The first projectile is launched at an angle _1=45^ + . The second projectile is launched at an angle _2=45^ - . Time of Flight Formula: The time of flight T for a projectile is given by: T= 2 vsin ⁡ g where v is the initial speed and g is the acceleration due to gravity. Ratio of Times of Flight: The ratio of the times of flights T_1T_2 can be determined as follows: T_1T_2= sin ⁡(45^ + ) sin ⁡(45^ - ) Applying Trigonometric Identities: Use the sine addition and subtraction formulas: sin ⁡(45^ + )= 1 2 ( cos ⁡ + sin ⁡ ) sin ⁡(45^ - )= 1 2 ( cos ⁡ - sin ⁡ ) Simplifying the Ratio: Substitute the expressions for sin ⁡(45^ + ) and sin ⁡(45^ - ) into the ratio: T_1T_2= 1 2 ( cos ⁡ + sin ⁡ ) 1 2 ( cos ⁡ - sin ⁡ )= cos ⁡ + sin ⁡ cos ⁡ - sin ⁡ Factor out the trigonometric functions to express in terms of tangent: T_1T_2= 1+ tan ⁡ 1- tan ⁡ Therefore, the ratio of the times of flights for the two projectiles is 1+ tan ⁡ 1- tan ⁡ .

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

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