Why is projectile motion parabolic?

Answer 1

Projectile motion is parabolic because horizontal velocity is typically constant and the object's vertical position is only affected by a constant acceleration (assuming constant drag, etc.).

In short, the fundamental motion of a projectile is parabolic because of its associated equation of motion.

#x(t) = 1/2 at^2 + v_i t + x_i#

is parabola-describing because it is quadratic.

But if you'd like, I can go into more detail about why this works by performing a little integration. Let's start 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#.

Let's move on 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#)

This is, of course, the equation of motion I mentioned earlier, so it should look familiar to you.

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.

If an object were dropped and fell straight down with no horizontal velocity, the path would not appear to be parabolic; instead, it would resemble a line, even though it is actually a parabola that has been compressed infinitely horizontally. In general, the more the parabola is compressed horizontally, the smaller the horizontal velocity.

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Answer 2

Lily Adams

Projectile motion is parabolic because the horizontal and vertical components of motion are independent, and gravity acts only vertically, causing the projectile to follow a curved path.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.