(B) The horizontal range $R$ is given by $R = u \cos \theta \cdot T$,where $u$ is the initial velocity and $\theta$ is the angle of projection.The tim

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$\tan^{-1} \left( \frac{gT^2}{2R} \right)$

(B) The horizontal range $R$ is given by $R = u \cos \theta \cdot T$,where $u$ is the initial velocity and $\theta$ is the angle of projection.The time of flight $T$ is given by $T = \frac{2u \sin \theta}{g}$,which implies $u \sin \theta = \frac{gT}{2}$.Dividing the vertical component by the horizontal component:$\tan \theta = \frac{u \sin \theta}{u \cos \theta} = \frac{gT/2}{R/T} = \frac{gT^2}{2R}$.Therefore,the angle of projection is $\theta = \tan^{-1} \left( \frac{gT^2}{2R} \right)$.

Class 11 Physics 2 Motion in a Plane Horizontal Projectile Motion

Medium

If the time of flight of a bullet over a horizontal range $R$ is $T \ s$,the inclination of the direction of projection to the horizontal is

A
$\sin^{-1} \left( \frac{gT^2}{R} \right)$
B
$\tan^{-1} \left( \frac{gT^2}{2R} \right)$
C
$\cos^{-1} \left( \frac{2gT^2}{2R} \right)$
D
$\cot^{-1} \left( \frac{R}{gT^2} \right)$

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2 Motion in a Plane

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Horizontal Projectile Motion

$A$ small ball is thrown at an angle $45^{\circ}$ to the horizontal with an initial velocity of $2 \sqrt{2} \ m/s$. The magnitude of the mean velocity averaged over the first $2 \ s$ is (take acceleration due to gravity $g = 10 \ m/s^2$). (in $m/s$)

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Three projectiles $A, B$ and $C$ are projected at an angle of $30^{\circ}, 45^{\circ}, 60^{\circ}$ respectively. If $R_{A}, R_{B}$ and $R_{C}$ are the ranges of $A, B$ and $C$ respectively,then (velocity of projection is the same for $A, B$ and $C$):

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Two objects are thrown up at angles of $45^{\circ}$ and $60^{\circ}$ respectively,with the horizontal. If both objects attain the same vertical height,then the ratio of the magnitude of velocities with which these are projected is .........

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$A$ body is thrown at an angle of $30^{\circ}$ to the horizontal with a velocity of $30\; m/s$. After $1\; s$,its velocity will be (in $m/s$) $\left(g=10\; m/s^{2}\right)$

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$A$ football player throws a ball with a velocity of $50 \ m/s$ at an angle of $30^{\circ}$ from the horizontal. The ball remains in the air for ...... $s$ $(g = 10 \ m/s^2)$.

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If the time of flight of a bullet over a horizontal range $R$ is $T \ s$,the inclination of the direction of projection to the horizontal is

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$A$ small ball is thrown at an angle $45^{\circ}$ to the horizontal with an initial velocity of $2 \sqrt{2} \ m/s$. The magnitude of the mean velocity averaged over the first $2 \ s$ is (take acceleration due to gravity $g = 10 \ m/s^2$). (in $m/s$)

Three projectiles $A, B$ and $C$ are projected at an angle of $30^{\circ}, 45^{\circ}, 60^{\circ}$ respectively. If $R_{A}, R_{B}$ and $R_{C}$ are the ranges of $A, B$ and $C$ respectively,then (velocity of projection is the same for $A, B$ and $C$):

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