We will cover following topics

Motion In A Plane

The motion in a plane can be analysed by writing the equations of motion for $x$ and $y$ axes. The following equations hold true:

v = u + a t

s = u t + \frac{1}{2} a t^{2}

v^{2} = u^{2} + 2 a s

For motion under gravity, $a_{y} = g$ , downwards.

Projectiles

Let the projectile be launched with an intitial velocity $v_{0} = v_{0 x} i + v_{0 y} j$ ,

where $v_{0 x} = v_{0} \cos θ$

and

v_{0 y} = v_{0} \sin θ

Acceleration: $a_{x} = 0$ and $a_{y} = - g$ .

Velocity: $v_{x} = v_{0} \cos (θ)$ , and $v_{y} = v_{0} \sin (θ) - g t$

Displacement: $x = v_{0} t \cos (θ)$ , and $y = v_{0} t \sin (θ) - \frac{1}{2} g t^{2}$ .

Eliminating $t$ , we get

$y = \tan (θ) \cdot x - \frac{g}{2 v_{0}^{2} \cos^{2} θ} \cdot x^{2}$ ,

which represents a parabolic path.

Time of Flight

Substituting $y = 0$ in displacement equation, we obtain time of flight,

$t = \frac{2 v_{0} \sin (θ)}{g}$

Maximum Elevation

Substituting $v_{y} = 0$ in velocity equation, we obtain maximum elevation,

$h = \frac{v_{0}^{2} \sin^{2} (θ)}{2 g}$

Range

$R = \frac{v_{0}^{2} \sin 2 θ}{g} = \frac{4 h}{t a n θ}$

PYQs

Projectiles

1) A particle projected from a given point on the ground just clears a wall of height $h$ at a distance from the point of projection. If the particle moves in a vertical plane and if the horizontal range is $R$ , find the elevation of the projection.

[2018, 10M]

2) A particle is projected from the base of a hill whose slope is that of a right circular cone, whose axis is vertical. The projectile gazes the vertex and strikes the hill again at a point on the base. If the semi vertical angle of the cone is $30^{\circ}$ , $h$ is height, determine the initial velocity on $u$ of the projection and its angle of projection.

[2015, 13M]

3) A particle is acted on by a force parallel to the axis of $y$ whose acceleration (always towards the axis of $x$ ) is $μ y - 2$ and when $y = a$ , it is projected parallel to the axis of $X$ with velocity $\sqrt{\frac{2 μ}{a}}$ . Find the parametric equation of the path of the particle. Here $μ$ is a constant.

[2014, 15M]

4) A projectile aimed at a mark which is in the horizontal plane through the point of projection, falls $x$ meter short of it when the angle of projection is $α$ and goes $y$ meter beyond when the angle of projection is $β$ . If the velocity of projection is assumed same in all cases, find the correct angle of projection.

[2011, 12M]

5) If $v_{1}, v_{2}, v_{3}$ are the velocities at three points $A, B, C$ of the path of a projectile, where the inclinations to the horizon are $α, α - β, α - 2 β$ , and if $t_{1}, t_{2}$ are the times of describing the arcs $A B, B C$ , respectively, prove that $v_{3} t_{1} = v_{1} t_{2}$ and $\frac{1}{v_{1}} + \frac{1}{v_{3}} = \frac{2 \cos β}{v_{2}}$ .

[2010, 12M]

6) A shot fired with a velocity $V$ at an elevation $α$ strikes a point $P$ in a horizontal plane through the point of projection. If the point $P$ is receding from the gun with $V$ , show that the elevation must be changed $θ$ , where $\sin 2 θ = \sin 2 α + \frac{2 v}{V} \sin θ$ .

[2009, 12M]

7) Prove that the velocity required to project a particle from a height $h$ to fall at a horizontal distance $a$ from a point of projection is at least equal to $\sqrt{g (\sqrt{a^{2} + h^{2} - h})}$ .

[2004, 15M]

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