It seems that the kinematic equations can all be derived from some simple definitions.

We’ll begin with our symbols:

= position (or distance)

= initial velocity

= final velocity

= acceleration

= time (or duration)

We have the definitions:

so

so

Now, for our purposes, we will regard and as constant, and as a function of .

Since is constant, we have (abusing notation a little)

Thus we have our first kinematic equation:

[1]

We will now integrate this (w.r.t. ):

so .

Using the boundary condition if , we have .

Thus we have our second kinematic equation:

[2]

Now, if we substitute [1] into [2] at , we obtain

and so we have our third kinematic equation (one not normally given)

[3]

If we substitute [1] into [2] differently, at , we obtain

so

and so we have our fourth kinematic equation

[4]

Finally, substituting [4] into [1] at t, we have

so

and so we have our fifth kinematic equation

[5]

To review what we have:

[1] or [¬s]: ; an equation without

[2] or [¬v]: ; an equation without

[3] or [¬u]: ; an equation without

[4] or [¬a]: ; an equation without

[5] or [¬t]: ; an equation without

(See also my post on parabolic trajectories.)

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