Constant acceleration equations
To derive the constant acceleration equations, we will need the following free fall equations.
v = v
0 + g × t
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 × t2
/
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
2d = 2v
0 × t + at
2 (equation a)
Replace t in equation a
2d =
v - v0
/
a
(2v
0 + v - v
0)
v
2 - v
02 = 2ad
v
2 = v
02 + 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 ×
t2
/
2
equation 2
Replace a in the latter equation 2
Replace a in the latter equation 2
d =
(v0 + 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 × t2
/
2
d = vt - at
2 +
a × t2
/
2
d = vt +
-2at2 + a × t2
/
2
d = vt +
-a × t2
/
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.
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