# For any given angle of launch and initial velocity prove from equations of motions that the trajectory of projectile motion is parabolic.

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### 1 Answer

Let the initial velocity of the object be v and the angle of launch be A. The initial velocity can be divided into a vertical component and a horizontal component that are given by `v*sin A` and `v*cos A ` respectively.

When the ball moves, the horizontal displacement at time t is equal to `x = v*cos A*t` . the vertical displacement on the other hand involves taking into account the acceleration of the object in a vertically downwards direction due to the force of gravitation between the Earth and the object. This gives `y = v*sin A*t - (1/2)*g*t^2`

Substitute `t = x/(v*cos A)` into this equation. This gives `y = v*sin A*x/(v*cos A) - (1/2)*g*(x/(v*cos A))^2`

=> `y = x*tan A - (g/(2*v^2*sin^2A))*x^2`

This is the equation of a parabola.