Deriving the Big 5 Equations

Equations are used as they are more versatile and easier to manipulate than graphs. In total there is a set of 5 equations which can be derived from a velocity vs. time graph. 


The first equation: constructed by taking the slope of the graph
a= rise/ run
a= v2-v1/ Δt       rearrange it:
v2= v1+ aΔt


Equation two: found by calculating the area under the graph (displacement)
Δd= ½ (v2+ v1)Δt


Now we can manipulate these two equations to create three more equations.


Substitute the expression for v2 into equation 2 to find equation 3:
Δd=½ (v1+aΔt+v1)Δt
Δd=½Δt (2v1+ aΔt)
Δd= v1Δt+½aΔt²


To find equation 4: first isolate v1 in equation 1, then substitute this in equation 2:
v1=v2-aΔt      subs. in equation 2:
Δd= ½ (v2+v2-aΔt) Δt
Δd= ½Δt(2v2-aΔt)
Δd= v2Δt- ½aΔt²


To find the final equation 5, isolate Δt in equation 1 and substitute it into equation 2:

Δt= v2-v1/a      subs. in equation 2:

Δd= ½ (v2+ v1) v2-v1/a

Δd= ½ (v2²- v1²) /a

2aΔd= v2²- v1²

v2²= v1²+ 2aΔd


These are the five fundamental equations in kinematics. They are applicable only in problems that have constant acceleration.