Trajectory of a projectile.

Question

From the definition of a parabola can we prove that the trajectory of a projectile is parabolic? And can this be proved by calculus?

Answer 1

I don't know if this can be proved using calculus but this a simple proof using equations of motion.

Let us assume that a projectile is being fired from the origin.

angle of projection = some $\theta$.

velocity of projection = $v$

therefore

-the upward component of velocity = $v\sin(\theta)$

using the equation $s=ut+\frac{at^2}2$, for the upward motion, we get

$$y=v\sin(\theta)t-\frac{gt^2}2\tag{1}$$

and as there is no acceleration in the horizontal direction $x=v(cos(\theta))t$.

threfore $t=\frac{x}{v\cos(\theta)}$. substituting the value of $t$ in equation $(1)$ we get $$y= x\tan(\theta)-\dfrac {gx^2}{2v^2\cos^2\theta},$$ which exactly resembles the equation of a parabola due to the presence of a squared term :

$y=ax-bx^2$. hope this helps!