Page History

...

\begin{large}\[ v_{f} = v_{i} + a(t_{f}-t_{i})\]\[x_{f} = x_{i} + \frac{1}{2}(v_{f}+v_{i})(t_{f}-t_{i})\]\[x_{f}=x_{i}+v_{i}(t_{f}-t_{i})+\frac{1}{2}a(t_{f}-t_{i})^{2}\]\[v_{f}^{2}=v_{i}^{2}+2a(x_{f}-x_{i})\]\end{large

}

It is clear from these equations that there are seven possible unknowns in a given problem involving motion between two points with constant acceleration:

1. initial time ( t_i )
2. final time ( t_f )
3. initial position ( x_i )
4. final position ( x_f )
5. initial velocity ( v_i )
6. final velocity ( v_f )
7. acceleration (constant) a

Looking at the four equations, you can see that each is specialized to deal with problems involving specific combinations of these unknowns.

Because

the

initial

position

and

initial

time

can

generally be

arbitrarily

chosen,

it

is

often useful to

rewrite

all

these

equations

in

terms

of

only

five variables

by

defining:

\begin{large}\[ \Delta x \equiv x_{f}-x_{i} \]\[\Delta t \equiv t_{f}-t_{i}\]\end{large}

then

each

of

the

equations

given

above

involves

exactly

four

unknowns.

Interestingly,

the

four

equations

represent

all

but

one

of

the

unique

combinations

of

four

variables

chosen

from

five

possible

unknowns.

Which

unique

combination

is

missing?

Can

you

derive

the

appropriate

"fifth

equation"?