Peoples Physics Book Version-2

http://www.ck12.org Chapter 4. One Dimensional Motion Version 2

4.3 Deriving the Kinematics Equations

The simplest case of one dimensional motion is an object at rest. A slightly more difficult problem is that of an
object moving at a constant velocity. Such an object’s position at timetis given by the familiard=rtformula, that
is, distance equals rate times time. In our language, this would be:

∆x=xf−xi=vt When velocity is constant [1]

If an object is undergoing an acceleration that changes with time, it is in general quite difficult to find its position and
velocity as a function of time. However, it’s always true that over a period of time∆taverage velocity and average
acceleration are given by:

vavg=

∆x

∆t

Always [2]

aavg=

∆v

∆t

Always [3],

In other words,

∆x=vavg∆t Always [4]

∆v=aavg∆t Always [5],

Therefore, finding an object’s position or velocity can be reduced to finding the average velocity or average accel-
eration, respectively. Usually, this is just as difficult as the problem mentioned above, but in one very common and
specific case —constant acceleration— these formulas are very useful. In this case,velocitychanges at a linear
rate with time, that is:

vf=at+vi When acceleration constant [6]

You should realize this is just another version of equation [1], which in fact describes anything changing at a linear
rate. Since the average of a linear function over some time is just the average of its endpoints (figure 1), we have:

vavg=

vf+vi

2

When acceleration constant [7]

Now, we

Start with equation [7] vavg=

vf+vi

2

Plug in equation [6] vavg=

at+vi+vi

2

And end up with vavg=

at

2

+vi

Finally, since∆x(t) =vavgt x(t) =xi+vit+^12 at^2 [8]

We have obtained the equations of uniformly, accelerated motion, also known as the

Big Three Equations

x(t) =x 0 +v 0 t+^12 at^2 [8]

v(t) =v 0 +at [6]

vf^2 =v 02 + 2 ax (Derivation left to reader) [9]