Equations of motion

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http://en.wikipedia.org/wiki/Equation_of_motion

In physics, equations of motion are equations that describe the behavior of a system (e.g., the motion of a particle under an influence of a force) as a function of time. Sometimes the term refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.

The equations that apply to bodies moving linearly (that is, one dimension) with uniform acceleration are presented below. They are often referred to as "UVAST", "SUVAT", "VUSAT", "VUATS" or "UVATS" equations, as the 5 variables they involve are often represented by those letters (s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time).

## Equations of uniformly accelerated linear motion

The body is considered between two instants in time: one "initial" point and one "current". Often, problems in kinematics deal with more than two instants, and several applications of the equations are required. If a is constant, a differential, a dt, may be integrated over an interval from 0 to Δt (Δt = tti), to obtain a linear relationship for velocity. Integration of the velocity yields a quadratic relationship for position at the end of the interval.

Note that each of the equations contains four of the five variables. Thus, in this situation it is sufficient to know three out of the five variables to calculate the remaining two.

## Classic version

The above equations are often written in the following form:

\begin{alignat}{3} & v&& = u+at\qquad & \text{(1)} \\ & s&& = \tfrac12(u+v)t\qquad & \text{(2)} \\ & s&& = ut + \tfrac12 at^2 \qquad & \text{(3)} \\ & v^2 && = u^2 + 2as\qquad & \text{(4)} \\ & s&& = vt - \tfrac12 at^2 \qquad & \text{(5)} \end{alignat}

By substituting (1) into (2), we can get (3), (4) and (5)

where

s = the distance between initial and final positions (displacement) (sometimes denoted R or x)
u = the initial velocity (speed in a given direction)
v = the final velocity
a = the constant acceleration
t = the time taken to move from the initial state to the final state

### Examples

Many examples in kinematics involve projectiles, for example a ball thrown upwards into the air.

Given initial speed u, one can calculate how high the ball will travel before it begins to fall.

The acceleration is local acceleration of gravity g. At this point one must remember that while these quantities appear to be scalars, the direction of displacement, speed and acceleration is important. They could in fact be considered as uni-directional vectors. Choosing s to measure up from the ground, the acceleration a must be in fact −g, since the force of gravity acts downwards and therefore also the acceleration on the ball due to it.

At the highest point, the ball will be at rest: therefore v = 0. Using the 4th equation, we have:

$s= \frac{v^2 - u^2}{-2g}.$

Substituting and cancelling minus signs gives:

$s = \frac{u^2}{2g}.$

### Extension

More complex versions of these equations can include a quantity Δs for the variation on displacement (ss0), s0 for the initial position of the body, and v0 for u for consistency.

$v = v_0 + at \,$
$s = s_0 + \tfrac{1}{2} (v_0 + v)t \,$
$s = s_0 + v_0 t + \tfrac{1}{2} at^2 \,$
$v^2 = v_0^2 + 2a\Delta s \,$
$s = s_0 + vt - \tfrac{1}{2} at^2 \,$

However a suitable choice of origin for the one-dimensional axis on which the body moves makes these more complex versions unnecessary.

## Equations of circular motion

The analogues of the above equations can be written for rotation:

$\omega=\omega_0+\alpha t\,$
$\phi=\tfrac12(\omega_0+\omega)t$
$\phi=\omega_0t+\tfrac12\alpha t^2\,$
$\omega^2=\omega_0^2+2\alpha\phi\,$
$\phi=\omega t-\tfrac12\alpha t^2\,$

where:

α is the angular acceleration
ω is the angular velocity
φ is the angular displacement
ω0 is the initial angular velocity.

## Derivation

### Motion equation 1

By definition of acceleration,

$a = \frac{\Delta v}{\Delta t}\quad\Rightarrow\quad\ a = \frac{v - u}{t}$

Hence

$at = v - u \,$
$v = u + at \,$

### Motion equation 2

By definition,

$\mathrm{ average\ velocity } = \frac{s}{t}$

Hence

$\begin{matrix} \frac{1}{2} \end{matrix} (u + v) = \frac{s}{t}$
$s = \begin{matrix} \frac{1}{2} \end{matrix} (u + v)t$

### Motion equation 3

$t = \frac{v - u}{a}$

Using Motion Equation 2, replace t with above

$s = \begin{matrix} \frac{1}{2} \end{matrix} (u + v) ( \frac{v - u}{a} )$
$2as = (u + v)(v - u) \,$
$2as = v^2 - u^2 \,$
$v^2 = u^2 + 2as \,$

### Motion equation 4

Using Motion Equation 1 to replace u in Motion Equation 3 gives

$s = vt - \begin{matrix} \frac{1}{2} \end{matrix} at^2$

Published - July 2009