Constant acceleration equations
To derive the constant acceleration equations, we will need the following free fall equations.
v = v
_{0} + g × t
d = v
_{0} × t +
g × t^{2}
/
2
Remember that g is the acceleration of gravity and it is a constant acceleration. It is not only gravity that can give a constant acceleration. A car can drive also with a constant acceleration. We can generalize the two equations above by replacing g with a.
a is this case is any constant acceleration.
v = v
_{0} + a × t
equation 1
d = v
_{0} × t +
a × t^{2}
/
2
equation 2
Derivation of more constant acceleration equations
We can use the equations above to get 3 more constant acceleration equations. To derive the constant acceleration equations, concepts of factoring, simplifying exponents, and fractions will be used.
Solve for t in equation 1
v = v
_{0} + a × t
v  v
_{0} = a × t
Rewrite equation 2
d =
2v_{0} × t + at^{2}
/
2
2d = 2v
_{0} × t + at
^{2} (equation a)
Replace t in equation a
2d = 2v _{0}
v  v_{0}
/
a

+ a
(v v_{0})^{2}
/
a^{2}

2d = 2v _{0}
v  v_{0}
/
a


2d = 2v _{0}
v  v_{0}
/
a

+ a
(v v_{0})^{2}
/
a^{2}

2d = 2v _{0}
v  v_{0}
/
a


2d =
v  v_{0}
/
a
(2v
_{0} + v  v
_{0})
2d =
v  v_{0}
/
a
(v + v
_{0})
2d =
v^{2}  v_{0}^{2}
/
a
v
^{2}  v
_{0}^{2} = 2ad
v
^{2} = v
_{0}^{2} + 2ad
equation 3
Solve for a in equation 1.
v = v
_{0} + a × t
v  v
_{0} = a × t
Equation 2 can also be rewritten as
d = v
_{0} × t + a ×
t^{2}
/
2
equation 2
Replace a in the latter equation 2
d = v _{0}t +
v  v_{0}
/
t


Replace a in the latter equation 2
d = v _{0}t +
v  v_{0}
/
t


d = v
_{0}t +
(v  v_{0})t
/
2
d =
2v_{0}t + vt  v_{0}t
/
2
d =
(v_{0} + v)t
/
2
equation 4
Finally, solve for v
_{0} in equation 1 and replace v
_{0} in equation 2.
v
_{0} = v  at
d = (v  at) × t +
a × t^{2}
/
2
d = vt  at
^{2} +
a × t^{2}
/
2
d = vt +
2at^{2} + a × t^{2}
/
2
d = vt +
a × t^{2}
/
2
equation 5
We then have five important constant acceleration equations
Any questions about how I derive the constant acceleration equations, send me an email.