system
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human
In regards to projectile motions, why can the initial velocity of the object found at one angle be used to solve for variables at different?

What I mean by that is if we have the initial velocity for an object being thrust outward at a specified angle, what proof do we have to say that we can use that initial velocity to theoretically solve for say, the distance the object will travel at a different launch angle or the time the object will be in the air for a different angle?
gpt
Newton's second law is a second-order differential equation because the highest derivative that occurs is the second. (Acceleration is the second derivative of position.) Generally, the order of the differential equation tells you how many arbitrary constants appear in the most general solution. Newton's second law is really a vector equation, which gives you three equations total (one each for the *x*-, *y*-, and *z*- directions). Since 3 x 2 = 6, the general solution has 6 arbitrary constants, 2 for each of the 3 directions.

This means that the motion of the projectile is completely determined by:

* *The form of Newton's second law.* In this case, we simply have **a** = -g**z** since the only force is gravity, which points down.
* *The initial velocity.* You are given **v***_0_* which is really 3 separate numbers (the components in each of the three directions).
* *The initial position.* You are given **r***_0_* which is really 3 separate numbers (the components in each of the three directions).

For projectile motion under uniform gravity, we usually confine the motion to two dimensions, the horizontal and vertical directions. (In other words, we assume that projectile moves only in the *x*- and *z*-directions.) So we really only need 4 total numbers, 2 numbers from the initial velocity and 2 numbers from the initial position. The initial position we usually just say is (0,0). That is, we put the origin wherever the projectile starts. So all that we need to fully describe the motion are the *x*- and *z*- components of the initial velocity. But those components can also be written as:

> v*_x_* = v cos(θ)

> v*_z_* = v sin(θ)

That is, giving the initial velocity is equivalent to giving the speed *v* and the launch angle θ. Hence the motion is completely determined by the speed and the launch angle.