Definitions

# Equations 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 SUVAT or VUSAT equations, as the 5 variables they involve are represented by those letters (s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time)

## Linear equations of motion under uniform acceleration

The body is considered at 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.
$v_f = v_i + aDelta t ,$
$d = begin\left\{matrix\right\} frac\left\{1\right\}\left\{2\right\} end\left\{matrix\right\} \left(v_i + v_f\right)Delta t$
$d = d_i + v_iDelta t + begin\left\{matrix\right\} frac\left\{1\right\}\left\{2\right\} end\left\{matrix\right\} aDelta t^2$
$v_f^2 = v_i^2 + 2ad ,$

where...
$d_i ,$ is the body's initial position
$v_i ,$ is the body's initial velocity
and its current state is described by:
$d ,$, the distance travelled from initial state (displacement)
$v_f ,$, The final velocity
$Delta t ,$, the time between the initial and current states
$a ,$, the constant acceleration, or in the case of bodies moving under the influence of gravity, g.

Note that each of the equations contains four of the five variables. When using the above formulae, 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:

$v = u+at ,$...(1)

$s = frac \left\{1\right\} \left\{2\right\}\left(u+v\right) t$...(2)

$s = ut + frac \left\{1\right\} \left\{2\right\} a t^2$...(3)

$v^2 = u^2 + 2 a s ,$...(4)

$s = vt - frac \left\{1\right\} \left\{2\right\} a t^2$...(5)

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

where

s = the distance travelled from the initial state to the final state (displacement)(note that s is sometimes replaced with 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:

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

Substituting and cancelling minus signs gives:

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

### Extension

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

$v = v_0 + at ,$
$s = s_0 + begin\left\{matrix\right\} frac\left\{1\right\}\left\{2\right\} end\left\{matrix\right\} \left(v_0 + v\right)t ,$
$s = s_0 + v_0 t + begin\left\{matrix\right\} frac\left\{1\right\}\left\{2\right\} end\left\{matrix\right\}\left\{at^2\right\} ,$
$\left(v\right)^2 = \left(v_0\right)^2 + 2a Delta R ,$
$s = s_0 + v t - begin\left\{matrix\right\} frac\left\{1\right\}\left\{2\right\} end\left\{matrix\right\}\left\{at^2\right\} ,$

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

## Rotational equations of motion

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

$omega = omega_0 + alpha t ,$
$phi = phi_0$
+ begin{matrix} frac{1}{2} end{matrix}(omega_0 + omega)t
$phi = phi_0 + omega_0 t + begin\left\{matrix\right\} frac\left\{1\right\}\left\{2\right\} end\left\{matrix\right\}alpha \left\{t^2\right\} ,$
$\left(omega\right)^2 = \left(omega_0\right)^2 + 2alpha Delta phi ,$
$phi = phi_0 + omega t - begin\left\{matrix\right\} frac\left\{1\right\}\left\{2\right\} end\left\{matrix\right\}alpha \left\{t^2\right\} ,$

where:

$alpha$ is the angular acceleration
$omega$ is the angular velocity
$phi$ is the angular displacement
$omega_0$ is the initial angular velocity
$phi_0$ is the initial angular displacement
$Delta phi$ is the variation on angular displacement ($phi$ - $phi_0$).

## Derivation

### Motion equation 1

By definition of acceleration,
$a = frac\left\{Delta v\right\}\left\{Delta t\right\}quadRightarrowquad a = frac\left\{v - u\right\}\left\{t\right\}$

Hence

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

### Motion equation 2

By definition,
$mathrm\left\{ average velocity \right\} = frac\left\{s\right\}\left\{t\right\}$

Hence

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

### Motion equation 3

$t = frac\left\{v - u\right\}\left\{a\right\}$
Using Motion Equation 2, replace t with above
$s = begin\left\{matrix\right\} frac\left\{1\right\}\left\{2\right\} end\left\{matrix\right\} \left(u + v\right) \left(frac\left\{v - u\right\}\left\{a\right\} \right)$
$2as = \left(u + v\right)\left(v - u\right) ,$
$2as = v^2 - u^2 ,$
$v^2 = u^2 + 2as ,$

### Motion equation 4

Using Motion Equation 1 to replace u in motion equation 3 gives
$s = vt - begin\left\{matrix\right\} frac\left\{1\right\}\left\{2\right\} end\left\{matrix\right\} at^2$