# Why is projectile motion parabolic?

May 19, 2014

Projectile motion is parabolic because the vertical position of the object is influenced only by a constant acceleration, (if constant drag etc. is also assumed) and also because horizontal velocity is generally constant.

Put simply, basic projectile motion is parabolic because its related equation of motion,

$x \left(t\right) = \frac{1}{2} a {t}^{2} + {v}_{i} t + {x}_{i}$

is quadratic, and therefore describes a parabola.

However, I can explain a bit more in-depth why this works, if you'd like, by doing a little integration. Starting with a constant acceleration,

$a = k$,

we can move on to velocity by integrating with respect to $t$. ($a = k$ is interpreted as being $a = k {t}^{0}$)

$v \left(t\right) = \int k \mathrm{dt} = k t + {v}_{i}$

The constant of integration here is interpreted to be initial velocity, so I've just named it ${v}_{i}$ instead of $C$.

Now, to position:

$x \left(t\right) = \int \left(k t + {v}_{i}\right) \mathrm{dt}$
$x \left(t\right) = \frac{1}{2} k {t}^{2} + {v}_{i} t + {x}_{i}$

Again, the constant of integration is interpreted in this case to be initial position. (denoted ${x}_{i}$)

Of course, this equation will probably look familiar to you. It's the equation of motion I described above.

Don't worry if you haven't learned about integration yet; the only thing you need to worry about is the power of $t$ as we move from acceleration to velocity to position. If $t$ was present in the initial $a = k$ equation, with a degree other than $0$, (in other words, if $a$ is changing over time) then after integration we would end up with a degree different from $2$. But since $a$ is constant, $t$ will always be squared in the equation for position, resulting in a parabola.

Since acceleration due to gravity is generally fairly constant at around $9.8 \frac{m}{s} ^ 2$, we can say that the trajectory of a projectile is parabolic.

A case where the path wouldn't appear to be parabolic is if an object were dropped, falling straight downwards, with no horizontal velocity. In this case the path looks more like a line, but it's actually a parabola which has been infinitely horizontally compressed. In general, the smaller horizontal velocity, the more the parabola is compressed horizontally.