Constant acceleration equations

To derive the constant acceleration equations, we will need the following free fall equations.

v = v0 + g × t   

d = v0 × t +
g × t2 / 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 = v0 + a × t      equation 1

d = v0 × t +
a × t2 / 2
     equation 2

 Derivation of more constant acceleration equations

We can use the equations above to get 3 more constant accelerations equations. To derive the constant accelerations equations, concepts of factoring, simplyfying exponents, and fractions will be used.

Solve for t in equation 1

v = v0 + a × t

v - v0 = a × t

t =
v - v0 / a

Rewrite equation 2

d =
2v0 × t + at2 / 2

2d = 2v0 × t + at2   (equation a)

Replace t in equation a

2d = 2v0 
v - v0 / a
 + a   
(v- v0)2 / a2

2d = 2v0 
v - v0 / a
  +    
(v- v0)2 / a
2d = 2v0 
v - v0 / a
 + a   
(v- v0)2 / a2

2d = 2v0 
v - v0 / a
  +    
(v- v0)2 / a

2d =
v - v0 / a
(2v0 + v - v0)

2d =
v - v0 / a
(v + v0)

2d =
v2 - v02 / a

v2 - v02 = 2ad

v2 = v02 + 2ad      equation 3

Solve for a in equation 1.

v = v0 + a × t

v - v0 = a × t

a =
v - v0 / t

Equation 2 can also be rewritten as

d = v0 × t + a ×
t2 / 2
     equation 2

Replace a in the latter equation 2

d = v0t   +  
v - v0 / t
  ×    
t2 / 2

Replace a in the latter equation 2

d = v0t   +  
v - v0 / t
  ×    
t2 / 2

d = v0t +
(v - v0)t / 2

d =
2v0t + vt - v0t / 2

d =
v0t + vt / 2

d =
(v0 + v)t / 2
    equation 4

Finally, solve for v0 in equation 1 and replace v0 in equation 2

v0 = v - at
d = (v - at) × t +
a × t2 / 2

d = vt - at2 +
a × t2 / 2

d = vt +
-2at2 + a × t2 / 2

d = vt +
-a × t2 / 2
    equation 5

In summary, the 5 constant accelerations equations are

v = v0 + a × t     missing quantity is d

d = v0 × t +
a × t2 / 2
    missing quantity is v

v2 = v02 + 2ad     missing quantity is t

d =
(v0 + v)t / 2
    missing quantity is a

d = vt +
-a × t2 / 2
    missing quantity is v0

Any questions about how I derive the constant acceleration equations, send me an email.


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