Kinematic equations

Four kinematic equations describe an object's motion:

1. First equation

From A as an initial point:

x(B) = x(A) + v(A) [t(B) - t(A)]+ (1/2) a [t(B) - t(A)]²     (1)

a (A → B) = (v(B) - V(A))/[t(B) - t(A)]     (2)

From B as an initial point:

x(A) = x(B) + v(B) [t(A) - t(B)] + (1/2) a [t(A) - t(B)]²     (3)

a (B → A) = (v_A - v_B)/[t(A) - t(B)] (4)

We have:
x(B) - x(A) = - [x(A) - x(B)]     (5)

a (A → B) = a (B → A)     (6)

(1) becomes:
x(B) - x(A) = -[ v(B) [t(A) - t(B)] + (1/2) a [t(A) - t(B)]²]
= - v(B) [t(A) - t(B)] - (1/2) a [t(A) - t(B)]²
or
x(B) = x(A) - v(B) [t(A) - t(B)] - (1/2) a [t(A) - t(B)]²
    (1')

Therefore:
x(B) can be written in two ways:
with (1)
x(B) = x(A) + v(A) [t(B) - t(A)]+ (1/2) a [t(B) - t(A)]²
    (7)
or with (1')
x(B) = x(A) - v(B) [t(A) - t(B)] - (1/2) a [t(A) - t(B)]²
= x(B) = x(A) + v(B) [t(B) - t(A)] - (1/2) a [t(B) - t(A)]²
    (8)

Particular cases

case 1:
A = O = Origin
x(B) = x
t(B) = t
v(B) = v_b
x(A) = x_o
v(A) = v_o
t(A) = 0
Therefore:
x = x_o + v_o t + (1/2) a t ²
or:
x = x_o + v_b t - (1/2) a t ²

case 2:
A = O = Origin
x(B) = x
t(B) = t
v(B) = v_b
x(A) = x_o = 0
v(A) = v_o
t(A) = 0
Therefore:
x = v_o t + (1/2) a t ²
or:
x = v_b t - (1/2) a t ²

x = v_o t + (1/2) a t ²
or
x = v_b t - (1/2) a t ²

2. Second equation

Adding (7) and (8), we obtain: x(B) = x(A) + v(A) [t(B) - t(A)]+ (1/2) a [t(B) - t(A)]² x(B) = x(A) + v(B) [t(B) - t(A)] - (1/2) a [t(B) - t(A)]² x(B) = x(A) + [v(A) + v(B)][t(B) - t(A)] /2     (9)

In the case of A is the origin:
x(B) = x V(B) = v
t(B) = t
t(A) = 0
V(A) = v_o x(A) = x_o = 0, we have:
x = x_o + (v + v_o)t /2
x = (v + v_o)t /2

x = (v + v_o)t /2

3. Third equation

The derivative of (7) gives the average velocity: [x(B) - x(A)]/[t(B) - t(A)] = v(A) + a [t(B) - t(A)] As the time interval [t(B) - t(A)] approaches zero, we have: v (B) = v(A) + a [t(B) - t(A)]    (10)

In the case of A is the origin:
V(B) = v
t(B) = t
t(A) = 0
V(A) = v_o, so
v = v_o + a t

v = v_o + a t

4. Forth equation

Substituting (10) in (1) gives:
x(B) = x(A) + v(A) [v (B) - v(A)]/a + (1/2) a ([v (B) - v(A)]/a)² =
x(B) = x(A) + v(A) [v (B) - v(A)]/a + (1/2) ([v (B) - v(A)]²/a
or
[x(B) - x(A)]a = [v (B) - v(A)] [v(A) + (1/2) ([v (B) - v(A)]]
= 2 a [x(B) - x(A)] = [v (B) - v(A)] [v(A) + v(B)] = v² (B) - v²(A)

In the case of A is the origin:
x(A) = x_o V(B) = v
V(A) = v_o, so
We have: v² - v_o² = 2 a [x - x_o]

v² - v_o² = 2 a (x - x_o)