PROJECTILE MOTION

Projectile motion is a form of motion in which an object or particle (called a projectile) is thrown near the earth's surface, and it moves along a curved path under the action of gravity only. The only force of significance that acts on the object is gravity, which acts downward to cause a downward acceleration. [image:https://i.imgur.com/7wAtGmJ.jpg] The motion can be split into two components. If '''u''' is the initial velocity at some angle θ, the horizontal component of the velocity, '''ucosθ''', is constant. The vertical component, '''usinθ''', accelerates due to gravity and changes with time. This acceleration is constant at every point in the flight. The combination of the two motions gives rise to the '''parabolic trajectory'''. =Example Cases= ==Object projected on a level ground== [image:https://i.imgur.com/DEcTFhd.jpg] The parabolic path is symmetrical so the time going up is equal to the time going down. To find the total time of flight, we only need to find the time going up or the time going down and multiply it by two. The horizontal range is simply the horizontal velocity multiplied by the time of flight. The time can be found using the initial vertical velocity, usinθ and the final at the peak of the flight which is zero. $$t=2 × \frac{u\sin\theta}{g}$$ The maximum height can be found using: $$ H = \frac{(u\sin\theta)^2}{2g}$$ And the range can be found using the time of flight and the constant horizontal velocity: $$R = u\cos\theta × t$$ '''NOTE''': The range formula can be expanded by substituting our expression for t above $$R = u \cos \theta × t = \frac{2u^2\sin\theta\cos\theta}{g}$$ Now since 2sinθcosθ = sin(2θ) $$R = \frac{u^2\sin2\theta}{g}$$