Projectile Motion questions | Projectile Motion jee questions

Q 1 :
The angle of projection for a projectile to have same horizontal range and maximum height is [2024]

$\tan^{- 1} (2)$
$\tan^{- 1} (\frac{1}{2})$
$\tan^{- 1} (4)$
$\tan^{- 1} (\frac{1}{4})$

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4

(C) $H_{m a x} = R a n g e$

$\frac{u^{2} \sin 2 θ}{g} \frac{u^{2} \sin^{2} θ}{2 g}$

$4 \sin θ \cos θ = \sin^{2} θ$

$\tan θ = 4 \Rightarrow θ = \tan^{- 1} 4$

Q 2 :
A particle moving in a circle of radius R with uniform speed takes time T to complete one revolution. If this particle is projected with the same speed at an angle $θ$ to the horizontal, the maximum height attained by it is equal to 4R. The angle of projection $θ$ is then given by [2024]

$\sin^{- 1} {[\frac{2 g T^{2}}{π^{2} R}]}^{\frac{1}{2}}$
$\sin^{- 1} {[\frac{π^{2} R}{2 g T^{2}}]}^{\frac{1}{2}}$
$\cos^{- 1} {[\frac{2 g T^{2}}{π^{2} R}]}^{\frac{1}{2}}$
$\cos^{- 1} {[\frac{π R}{2 g T^{2}}]}^{\frac{1}{2}}$

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(A) $\frac{2 π R}{T} = v$

Maximum height $H = \frac{v^{2} \sin^{2} θ}{2 g}$

$4 R = \frac{4 π^{2} R^{2}}{T^{2} 2 g} \sin^{2} θ$

$\sin θ = \sqrt{\frac{2 g T^{2}}{π^{2} R}} \Rightarrow θ = \sin^{- 1} {(\frac{2 g T^{2}}{π^{2} R})}^{\frac{1}{2}}$

Q 3 :
The maximum height reached by a projectile is 64 m. If the initial velocity is halved, the new maximum height of the projectile is _____ m. [2024]

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(16) $H_{\max} = 64 m$

$H_{\max} = \frac{u^{2} \sin^{2} θ}{2 g} \Rightarrow \frac{H_{1 \max}}{H_{2 \max}} = \frac{u_{1}^{2}}{u_{2}^{2}}$

$\Rightarrow \frac{64}{H_{2 \max}} = \frac{u^{2}}{{(\frac{u}{2})}^{2}} \Rightarrow H_{2 \max} = 16 m$

Q 4 :
A body of mass M thrown horizontally with velocity $v$ from the top of the tower of height H touches the ground at 100 m from the foot of the tower. A body of mass 2M thrown at a velocity $\frac{v}{2}$ from the top of the tower of height 4H will touch the ground at a distance of _____ m. [2024]

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(100)

Q 5 :
Projectiles A and B are thrown at angles of 45° and 60° with the vertical, respectively, from the top of a 400 m high tower. If their ranges and times of flight are the same, the ratio of their speeds of projection $v_{A} : v_{B}$ is [2024].

$1 : \sqrt{3}$
$\sqrt{2} : 1$
$1 : 2$
$1 : \sqrt{2}$

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(4)

$R_{A} = R_{B}$

$(V_{A} \cos 45 °) T = (V_{B} \cos 60 °) T$

$\frac{V_{A}}{\sqrt{2}} = \frac{V_{B}}{2}$

$\frac{V_{A}}{V_{B}} = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$

Q 6 :
A ball rolls off the top of a stairway with horizontal velocity u. The steps are 0.1 m high and 0.1 m wide. The minimum velocity u with which that ball just hits step 5 of the stairway will be $\sqrt{x} m s^{- 1}$ , where $x$ = _____.

[Use g = 10 $m / s^{2}$ ]. [2024]

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(2)

The ball needs to just cross 4 steps to just hit the 5th step.

Therefore, horizontal range (R) = 0.4 m.

$R = u \cdot t$

In vertical direction, $h = \frac{1}{2} g t^{2}$

$0.4 = \frac{1}{2} g t^{2} = \frac{1}{2} g {(\frac{0.4}{u})}^{2}$

$u^{2} = 2 \Rightarrow u = \sqrt{2} m/s$

Therefore, $x = 2$

Q 7 :
A body starts falling freely from height $H$ and hits an inclined plane in its path at height $h$ . As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of $H / h$ for which the body will take the maximum time to reach the ground is ______. [2024]

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(2)

Total time of flight = T

$T = \sqrt{\frac{2 h}{g}} + \sqrt{\frac{2 (H - h)}{g}}$

For maximum time $= \frac{d T}{d h} = 0$

$\sqrt{\frac{2}{g}} (\frac{- 1}{2 \sqrt{H - h}} + \frac{1}{2 \sqrt{h}}) = 0$

$\sqrt{H - h} = \sqrt{h}$

$h = \frac{H}{2} \Rightarrow \frac{H}{h} = 2$

Topic Question Set

Q 1 :
The angle of projection for a projectile to have same horizontal range and maximum height is [2024]

Q 2 :
A particle moving in a circle of radius R with uniform speed takes time T to complete one revolution. If this particle is projected with the same speed at an angle $θ$ to the horizontal, the maximum height attained by it is equal to 4R. The angle of projection $θ$ is then given by [2024]

Q 3 :
The maximum height reached by a projectile is 64 m. If the initial velocity is halved, the new maximum height of the projectile is _____ m. [2024]

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Q 5 :
Projectiles A and B are thrown at angles of 45° and 60° with the vertical, respectively, from the top of a 400 m high tower. If their ranges and times of flight are the same, the ratio of their speeds of projection $v_{A} : v_{B}$ is [2024].

Q 6 :
A ball rolls off the top of a stairway with horizontal velocity u. The steps are 0.1 m high and 0.1 m wide. The minimum velocity u with which that ball just hits step 5 of the stairway will be $\sqrt{x} m s^{- 1}$ , where $x$ = _____.

[Use g = 10 $m / s^{2}$ ]. [2024]

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Topic Question Set

Q 1 : The angle of projection for a projectile to have same horizontal range and maximum height is [2024]

Q 2 : A particle moving in a circle of radius R with uniform speed takes time T to complete one revolution. If this particle is projected with the same speed at an angle θ to the horizontal, the maximum height attained by it is equal to 4R. The angle of projection θ is then given by [2024]

Q 3 : The maximum height reached by a projectile is 64 m. If the initial velocity is halved, the new maximum height of the projectile is _____ m. [2024]

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Q 4 : A body of mass M thrown horizontally with velocity v from the top of the tower of height H touches the ground at 100 m from the foot of the tower. A body of mass 2M thrown at a velocity v2from the top of the tower of height 4H will touch the ground at a distance of _____ m. [2024]

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Q 5 : Projectiles A and B are thrown at angles of 45° and 60° with the vertical, respectively, from the top of a 400 m high tower. If their ranges and times of flight are the same, the ratio of their speeds of projection vA:vB is [2024].

Q 6 : A ball rolls off the top of a stairway with horizontal velocity u. The steps are 0.1 m high and 0.1 m wide. The minimum velocity u with which that ball just hits step 5 of the stairway will be xms-1, where x = _____. [Use g = 10 m/s2]. [2024]

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Q 1 :
The angle of projection for a projectile to have same horizontal range and maximum height is [2024]

Q 2 :
A particle moving in a circle of radius R with uniform speed takes time T to complete one revolution. If this particle is projected with the same speed at an angle $θ$ to the horizontal, the maximum height attained by it is equal to 4R. The angle of projection $θ$ is then given by [2024]

Q 3 :
The maximum height reached by a projectile is 64 m. If the initial velocity is halved, the new maximum height of the projectile is _____ m. [2024]

Q 5 :
Projectiles A and B are thrown at angles of 45° and 60° with the vertical, respectively, from the top of a 400 m high tower. If their ranges and times of flight are the same, the ratio of their speeds of projection $v_{A} : v_{B}$ is [2024].

Q 6 :
A ball rolls off the top of a stairway with horizontal velocity u. The steps are 0.1 m high and 0.1 m wide. The minimum velocity u with which that ball just hits step 5 of the stairway will be $\sqrt{x} m s^{- 1}$ , where $x$ = _____.

[Use g = 10 $m / s^{2}$ ]. [2024]