Projectile motion is a 2D motion. Common examples are football hit by a player, bullet fired from a gun, catapult, water coming out of hose pipe, mud particles thrown away from wheels of a vehicle, a high-jump athlete etc.
Note that aeroplane/spacecraft are NOT projectiles, because the propulsive force can be adjusted after their launch. In all previous cases, once the object is launched, its motion is purely under gravitational influence.
Let the velocity of projection be and the angle of projection be . The motion can be resolved into 2 motions:
a) along horizontal direction : This is a uniform velocity motion,
b) along vertical direction : This is a uniformly accelerated motion, with initial velocity and acceleration
This resolution makes the analysis simpler. We find the following factors:
1) Max. height reached
At the top, vertical velocity . Using ,
2) Time of flight
Time required to reach the top will be . Using ,
3) Horizontal range
This is the distance between the point of launch and the point where the projectile lands. Using ,
The maximum values of and can be obtained by substituting maximum values of trigonometric functions, and .
4) Equation of trajectory
On eliminating , we get,
This is of the form , which is a parabola.
Analysis of motion, when the ground is not horizontal, but inclined at an angle with the horizontal
There are 2 ways of analyzing this motion:
I) Resolving the acceleration vector along the ground and perpendicular to the ground :
In this case, both motions are uniformly accelerated, and
II) Considering the equation of ground as
All quantities can be found by considering equations of parabola and straight line.
1) Time of flight,
2) Range along plane,
For maximum range,
3) Max. height reached (perpendicular to ground),
Few more important points
I) Throughout this analysis, air resistance, effect of wind and effect of rotation of earth are neglected.
II) For same range, there are 2 angles of projections possible. These angles are complementary, i.e.
III) At any point, the radius of curvature is . is the angle made by velocity at that point with horizontal.
IV) Minimum radius of curvature is at the max. height