Question 15: Motion in 2D & Vectors

Question

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

Accepted Answer

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 $	heta_1 = 45^\circ + \alpha$.
The second projectile is launched at an angle $	heta_2 = 45^\circ - \alpha$.
Time of Flight Formula:
The time of flight $T$ for a projectile is given by:
$      T = \frac{2v \sin 	heta}{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 $\frac{T_1}{T_2}$ can be determined as follows:
$      \frac{T_1}{T_2} = \frac{\sin(45^\circ + \alpha)}{\sin(45^\circ - \alpha)}      $
Applying Trigonometric Identities:
Use the sine addition and subtraction formulas:
$      \sin(45^\circ + \alpha) = \frac{1}{\sqrt{2}}(\cos \alpha + \sin \alpha)      $
$      \sin(45^\circ - \alpha) = \frac{1}{\sqrt{2}}(\cos \alpha - \sin \alpha)      $
Simplifying the Ratio:
Substitute the expressions for $\sin(45^\circ + \alpha)$ and $\sin(45^\circ - \alpha)$ into the ratio:
$      \frac{T_1}{T_2} = \frac{\frac{1}{\sqrt{2}} (\cos \alpha + \sin \alpha)}{\frac{1}{\sqrt{2}} (\cos \alpha - \sin \alpha)} = \frac{\cos \alpha + \sin \alpha}{\cos \alpha - \sin \alpha}      $
Factor out the trigonometric functions to express in terms of tangent:
$      \frac{T_1}{T_2} = \frac{1 + 	an \alpha}{1 - 	an \alpha}      $
Therefore, the ratio of the times of flights for the two projectiles is $\frac{1 + 	an \alpha}{1 - 	an \alpha}$.

Answer

$\frac{1+	an \alpha}{1-	an \alpha}$

Answer

$\frac{1+\sin 2 \alpha}{1-\sin 2 \alpha}$

Answer

$\frac{1-	an \alpha}{1+	an \alpha}$

Answer

1

Motion in 2D & Vectors

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

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