Projectile Motion questions | Projectile Motion jee questions

Q 11 :
The angle of projection of a particle is measured from the vertical axis as $ϕ$ and the maximum height reached by the particle is $h_{m}$ . Here $h_{m}$ as function of $ϕ$ can be represented as [2025]

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(3)

$h_{m} = \frac{u^{2} \sin^{2} θ}{2 g}, θ = 90 ° - ϕ$

$\Rightarrow h_{m} = \frac{u^{2} \cos^{2} ϕ}{2 g}$

From $ϕ = 0 to ϕ = 90$

$h_{m}$ decreases with $\cos^{2}$ function.

Q 12 :
A particle is projected with velocity $u$ so that its horizontal range is three times the maximum height attained by it. The horizontal range of the projectile is given as $\frac{n u^{2}}{25 g}$ , where value of n is : (Given 'g' is the acceleration due to gravity). [2025]

6
18
12
24

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(4)

Range R = $3 H_{\max}$

$\frac{u^{2} \sin 2 θ}{g} = \frac{3 u^{2} \sin^{2} θ}{2 g}$

$2 \sin θ \cos θ = \frac{3}{2} \sin^{2} θ$

$\tan θ = \frac{4}{3} \Rightarrow θ = 53 °$

$R = \frac{u^{2} (2 \times \frac{3}{5} \times \frac{4}{5})}{g} = \frac{24 u^{2}}{25 g}$

Q 13 :
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]

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

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(4)

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

$T_{1} = \frac{2 u \sin (45 + α)}{g}$

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

$T_{2} = \frac{2 u \sin (45 - α)}{g}$

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

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

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

Q 14 :
A helicopter flying horizontally with a speed of 360 km/h at an altitude of 2 km, drops an object at an instant. The object hits the ground at a point 'O', 20 s after it is dropped. Displacement of 'O' from the position of helicopter where the object was released is:

(Use acceleration due to gravity g = 10 $m / s^{2}$ and neglect air resistance) [2025]

$2 \sqrt{5}$ km
4 km
7.2 km
$2 \sqrt{2}$ km

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(4)

$u = 360 \times \frac{5}{18} = 100 m / s$

$x = u \times t = 100 \times 20 = 2 \times 10^{3} m = 2 km$

$t = \sqrt{\frac{2 H}{g}} \Rightarrow H = \frac{t^{2} g}{2}$

$H = \frac{400 \times 10}{2} = 2000 m = 2 km$

Displacement, $D = \sqrt{x^{2} + H^{2}} = 2 \sqrt{2} km$ .

Q 15 :
Two balls with same mass and initial velocity, are projected at different angles in such a way that maximum height reached by first ball is 8 times higher than that of the second ball. $T_{1}$ and $T_{2}$ are the total flying times of first and second ball, respectively, then the ratio of $T_{1}$ and $T_{2}$ is : [2025]

$2 \sqrt{2} : 1$
2 : 1
$\sqrt{2} : 1$
4:1

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(1)

Given, ${(H)}_{1} = 8 \times {(H)}_{2}$

$H_{1} = \frac{u^{2} \sin^{2} θ_{1}}{2 g} and H_{2} = \frac{u^{2} \sin^{2} θ_{2}}{2 g}$

$\frac{u^{2} \sin^{2} θ_{1}}{2 g} = 8 \times \frac{u^{2} \sin^{2} θ_{2}}{2 g}$

$\sin θ_{1} = 2 \sqrt{2} \sin θ_{2} \Rightarrow \frac{\sin θ_{1}}{\sin θ_{2}} = 2 \sqrt{2}$

$T_{1} = \frac{2 u \sin θ_{1}}{g} and T_{2} = \frac{2 u \sin θ_{2}}{g}$

$\Rightarrow \frac{T_{1}}{T_{2}} = \frac{\sin θ_{1}}{\sin θ_{2}} = 2 \sqrt{2}$

Q 16 :
A particle is projected at an angle of 30° from horizontal at a speed of 60 m/s. The height traversed by the particle in the first second is $h_{0}$ and height traversed in the last second, before it reaches the maximum height, is $h_{1}$ . The ratio $h_{0}$ : $h_{1}$ is ___________ [Take, g = 10 $m / s^{2}$ ] [2025]

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(5)

Time to reach maximum height,

$t_{H_{\max}} = \frac{u \sin θ}{g} = \frac{60}{10} \times \frac{1}{2} = 3 s$

Height in $1^{st}$ second $S_{1} = 30 \times 1 - \frac{1}{2} \times 10 \times 1 = 25 m$

Height in last second $S_{3} = 30 + (\frac{- 10}{2}) \times (2 \times 3 - 1) = 5 m$

$\Rightarrow \frac{S_{1}}{S_{3}} = \frac{25}{5} = 5$

Topic Question Set

Q 11 :
The angle of projection of a particle is measured from the vertical axis as $ϕ$ and the maximum height reached by the particle is $h_{m}$ . Here $h_{m}$ as function of $ϕ$ can be represented as [2025]

Q 12 :
A particle is projected with velocity $u$ so that its horizontal range is three times the maximum height attained by it. The horizontal range of the projectile is given as $\frac{n u^{2}}{25 g}$ , where value of n is : (Given 'g' is the acceleration due to gravity). [2025]

Q 13 :
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]

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Topic Question Set

Q 11 : The angle of projection of a particle is measured from the vertical axis as ϕ and the maximum height reached by the particle is hm. Here hm as function of ϕ can be represented as [2025]

Q 12 : A particle is projected with velocity u so that its horizontal range is three times the maximum height attained by it. The horizontal range of the projectile is given as nu225g, where value of n is : (Given 'g' is the acceleration due to gravity). [2025]

Q 13 : 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]

Q 16 : A particle is projected at an angle of 30° from horizontal at a speed of 60 m/s. The height traversed by the particle in the first second is h0 and height traversed in the last second, before it reaches the maximum height, is h1. The ratio h0 : h1 is ___________ [Take, g = 10 m/s2] [2025]

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Q 11 :
The angle of projection of a particle is measured from the vertical axis as $ϕ$ and the maximum height reached by the particle is $h_{m}$ . Here $h_{m}$ as function of $ϕ$ can be represented as [2025]

Q 12 :
A particle is projected with velocity $u$ so that its horizontal range is three times the maximum height attained by it. The horizontal range of the projectile is given as $\frac{n u^{2}}{25 g}$ , where value of n is : (Given 'g' is the acceleration due to gravity). [2025]

Q 13 :
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]