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+12at2 v2=u2+2asFor motion under gravity, ay=g, downwards.
Projectiles
Let the projectile be launched with an intitial velocity v0=v0xi+v0yj,
where v0x=v0cosθ
and
v0y=v0sinθAcceleration: ax=0 and ay=−g.
Velocity: vx=v0cos(θ), and vy=v0sin(θ)−gt
Displacement: x=v0tcos(θ), and y=v0tsin(θ)−12gt2.
Eliminating t, we get
y=tan(θ)⋅x−g2v20cos2θ⋅x2,
which represents a parabolic path.
Maximum Elevation
Substituting vy=0 in velocity equation, we obtain maximum elevation,
h=v20sin2(θ)2g
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∘, 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 μy−2 and when y=a, it is projected parallel to the axis of X with velocity √2μa. 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 v1,v2,v3 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−2β, and if t1,t2 are the times of describing the arcs AB,BC, respectively, prove that v3t1=v1t2 and 1v1+1v3=2cosβv2.
[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 sin2θ=sin2α+2vVsinθ.
[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 √g(√a2+h2−h).
[2004, 15M]