Projectiles
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:
\[v=u+at\] \[s=ut+ \dfrac{1}{2}at^2\] \[v^2=u^2+2as\]For motion under gravity, \(a_y=g\), downwards.
Projectiles
Let the projectile be launched with an intitial velocity \(v_0=v_{0 x} \mathbf{i}+v_{0 y} \mathbf{j}\),
where \(v_{0 x}=v_{0} \cos \theta\)
and
\[v_{0 y}=v_{0} \sin \theta\]Acceleration: \(a_x=0\) and \(a_y=-g\).
Velocity: \(v_{x}=v_{0} \cos (\theta)\), and \(v_{y}=v_{0} \sin (\theta)-g t\)
Displacement: \(x=v_{0} t \cos (\theta)\), and \(y=v_{0} t \sin (\theta)-\dfrac{1}{2} g t^{2}\).
Eliminating \(t\), we get
\(y=\tan (\theta) \cdot x-\dfrac{g}{2 v_{0}^{2} \cos ^{2} \theta} \cdot x^{2}\),
which represents a parabolic path.
Time of Flight
Substituting \(y=0\) in displacement equation, we obtain time of flight,
\(t=\dfrac{2 v_{0} \sin (\theta)}{g}\)
Maximum Elevation
Substituting \(v_y=0\) in velocity equation, we obtain maximum elevation,
\(h=\dfrac{v_{0}^{2} \sin ^{2}(\theta)}{2 g}\)
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 \(\mu y-2\) and when \(y=a\), it is projected parallel to the axis of \(X\) with velocity \(\sqrt{\dfrac{2 \mu}{a}}\). Find the parametric equation of the path of the particle. Here \(\mu\) 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 \(\alpha\) and goes \(y\) meter beyond when the angle of projection is \(\beta\). 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 \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) of the path of a projectile, where the inclinations to the horizon are \(\alpha, \alpha-\beta, \alpha-2 \beta\), and if \(\mathrm{t}_{1}, \mathrm{t}_{2}\) are the times of describing the arcs \(\mathrm{AB}, \mathrm{BC}\), respectively, prove that \(v_{3} t_{1}=v_{1} t_{2}\) and \(\dfrac{1}{v_{1}}+\dfrac{1}{v_{3}}=\dfrac{2 \cos \beta}{v_{2}}\).
[2010, 12M]
6) A shot fired with a velocity \(V\) at an elevation \(\alpha\) 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 \(\theta\), where \(\sin 2 \theta=\sin 2 \alpha+\dfrac{2 v}{V} \sin \theta\).
[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 \(\sqrt{g\left(\sqrt{a^{2}+h^{2}-h}\right)}\).
[2004, 15M]