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titleFour Useful Equations
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Latex
\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}
{latex}{center}
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 ( ti )
  2. final time ( tf )
  3. initial position ( xi )
  4. final position ( xf )
  5. initial velocity ( vi )
  6. final velocity ( vf )
  7. acceleration (constant) a
 Note
Remember that you will usually have the freedom to define the initial time and the initial position by setting up a coordinate system.
Looking at the four equations, you can see that each is specialized to deal with problems involving specific combinations of these unknowns.
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titleAn Exercise in Derivation
Because 
 
 the 
 
 initial 
 
 position 
 
 and 
 
 initial 
 
 time 
 
 can 
 
generally be 
 
 arbitrarily 
 
 chosen, 
 
 it 
 
 is 
 possible 
often useful to 
 
 rewrite 
 
 all 
 
 these 
 
 equations 
 
 in 
 
 terms 
 
 of 
 
 only 
 5 
five variables 
 
 by 
 
 defining:

{latex}

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


 If 
 
 you 
 
 replace 
 
 the 
 
 initial 
 
 and 
 
 final 
 
 positions 
 
 and 
 
 times 
 
 with 
 
 these 
 
 "deltas", 
 
 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"?