Why is projectile motion parabolic?

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(t) = 1/2 at^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 = kt^0#)

#v(t) = int k dt = kt + 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(t) = int (kt + v_i) dt#
#x(t) = 1/2 kt^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 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.