Projectile Motion
Projectile refers to an object that is in flight after being thrown or projected. Only acceleration in projectile motion is gravity.
We are taking reference axis, any movement ,acceleration or velocity towards right is taken as positive(+ve) and towards left direction is negative(-ve) and any movement, velocity & acceleration towards upper direction is positive and in downwards direction is negative(-ve).
We will derive the equations for Time Period of projectile(T), maximum vertical displacement or height of projectile (H), Horizontal Displacement or Range(R).
Let the initial speed with which a projectile is thrown be u, and velocity at landing v.The angle θ be the angle the projectile object makes with the ground. The horizontal velocity component initially will be u cos θ and vertical initial velocity will be u sin θ.
We will recall the 3 equations of motion :-
i) v=u+at
ii) v²-u²=2as
iii) s=ut+(1/2) at²
We know that when we throw the ball in air while fielding directing it upwards it ascents upto a certain height & then it starts to comes back towards ground the reason is gravitational acceleration (g), the vertical speed on top most point of projectile will be 0.
using v=u+at
vᵧ=uᵧ+at
0=u sin θ+(-g)t
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t=u sin θ/g
but t will represent only half the time period as it will take equal time to come back to ground. Hence T =2*t
T=2 u sin θ/g
There is no acceleration in horizontal direction so horizontal displacement will be (using velocity=displacement/time)
u cos θ=R/T
R=u cos θ * (2 u sin θ / g)
R= u² (2 sin θ cos θ) /g
R= (u² sin 2θ) /g
For H or maximum vertical displacement using equation v²-u²=2as
vᵧ²-uᵧ²=2as
0-(u sin θ)²=2*(-g)*H
2*g*H=u² (sin θ)²
H=u² sin² θ/2 g
Position of projectile at any time t
x=u cos θ * t
y =u (sin θ) t - (1/2)gt²
