The equations that quantitatively describe uniformly accelerated rectilinear motion are derived below.
Let:
\( a \) = constant acceleration
\( u \) = initial velocity at time \( t_1 \)
\( v \) = final velocity at time \( t_2 \)
\( t = t_2 - t_1 \) = time interval
\( x \) = displacement during time interval \( t \)
\( x_0 \) = initial position at time \( t_1 \)
From the definition of constant acceleration:
\[ a = \frac{v - u}{t} \]
The kinematic equations for uniform acceleration are:
| \[ v = at + u \] | (1) - from acceleration definition |
| \[ x = \frac{1}{2}at^2 + ut + x_0 \] | (2) - from integration of velocity |
| \[ x = \frac{1}{2}(u + v)t + x_0 \] | (3) - using average velocity |
| \[ v^2 = u^2 + 2a(x - x_0) \] | (4) - time-independent relation |
| \[ v = at + u \] | (1) |
| \[ x = \frac{1}{2}at^2 + ut \] | (2) |
| \[ x = \frac{1}{2}(u + v)t \] | (3) |
| \[ v^2 = u^2 + 2ax \] | (4) |
| \[ v = at \] | (1) |
| \[ x = \frac{1}{2}at^2 \] | (2) |
| \[ x = \frac{1}{2}vt \] | (3) |
| \[ v^2 = 2ax \] | (4) |