Prove that the path of a projectile is a parabola. - Vidyadip

Question

Prove that the path of a projectile is a parabola.

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Answer

Consider a projectile thrown at an angle $\theta$ with a velocity u. The components of velocity horizontal and vertical are $\text{u}\cos\theta$ and $\text{u}\sin\theta.$ After time t, the horizontal displacement $\text{x}=\text{u}\cos\theta\text{t}$

The vertical displacement,

$\text{y}=\text{u}\sin\theta\text{ t}-\frac{1}{2}\text{gt}^2$

$\therefore\ \text{y}=\text{u}\sin\theta.\Big(\frac{\text{x}}{\text{u}\cos\theta}\Big)-\frac{1}{2}\text{g}\Big(\frac{\text{x}}{\text{u}\cos\theta}\Big)^2$

$\text{y}\propto\text{x}^2,$ the path of a projectile is a parabola.

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