is parabolic because the vertical position of the object is influenced only by a constant , (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 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.

is parabolic because the vertical position of the object is influenced only by a constant , (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 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.