show that path of a projectile is a parabola.

Suppose a projectile of mass ‘m’ is projected with velocity ‘u’ at an angle θ with the ground.

The horizontal component of the initial velocity is, u_{x }= u cosθ [this remains constant]

The horizontal displacement at any time ‘t’ is, X = u_{x }t = ut cosθ

=> t = X/(u cosθ)

The vertical component is given by, u_{y }= u sinθ

The vertical displacement at any time ‘t’ is, Y = u_{y }t – ½ gt ^{2 }= ut sinθ – ½ gt^{2}

=> Y = u[X/(u cosθ)]sinθ – ½ g[X/(u cosθ)]^{2}

=> Y = X tanθ – gX^{2 }/[2u^{2 }cos^{2 }θ]

Which is the equation of a parabola. So, the path of the projectile is parabolic.

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