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.
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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