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I've only learned about to use kinematics equation when solving projectile problems but today i came across the following equations. where does they come from?

  • Distance travelled
  • Time of flight
  • Angle of reach
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At the introductory level they come from (1) an assumption of no air resistance (2) an assumption of constant gravitation (3) the independence of the orthogonal components (i.e. the x motion does not effect the y motion and vice versa) and (4) the basic kinematics of constant acceleration motion. But as stated this question would take two hours in the lecture hall, please be more specific. –  dmckee Feb 27 '11 at 16:20
    
They come from Newton's laws of motion. Thats all you need, together for some assumptions about the air, gravity and the earth being locally flat. –  ja72 Jun 1 '11 at 23:13
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2 Answers

All the basic kinematic equation you've learned come from one basic equation,

$$\frac{\textrm{d}^2\vec{x}}{\textrm{d}t^2} = \vec{g}$$

This is a vector-valued second order linear differential equation. $ \vec{x}$ is the position of the projectile. $m$ is its mass. $\vec{g}$ is the acceleration due to gravity.

The equation can be read in simplified form as

$$\frac{\vec{F}}{m} = \vec{a}$$

This is called Newton's Second Law. $\vec{F}$ is the force on an object, which for a projectile near Earth's surface is $m\vec{g}$, ignoring all effects besides simple gravity (air resistance is the big one). $\vec{a}$ is the acceleration, which by definition is $\frac{\textrm{d}^2\vec{x}}{\textrm{d}t^2}$.

The kinematic equations that tell you, for example, the distance traveled by a projectile or its speed as a function of height can all be derived from this differential equation using calculus and algebra.

The differential equation itself is a postulate of Newtonian mechanics. In this sense it doesn't come from anywhere; it is simply assumed, along with the assumption that the force of gravity is constant.

However, with more advanced physics it is possible to explain Newtonian mechanics in terms of other theories. This simplest example is that the assumption that the force from gravity is $m\vec{g}$ can be understood in terms of Newton's theory of gravity. What we see is that the simple gravity law is only an approximation, but a good one.

Similarly, as we bring in more and more considerations, we tend to find that almost everything we've done in the beginning is only an approximation. The kinematic equations aren't fundamental laws of nature. They're mathematical consequences of a simple differential equation that is roughly true for everyday things.

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As dmckee, pointed out in his comment, these equations come from (or derived from) some basic assumptions. To understand how we obtain these equations from the basic assumptions, you will need to read derivations of these equations.

Watch this video for basic understanding. Then watch Khan Academy lectures on projectile motion. (only 3 lectures).

May the force be with you!

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