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 equations, as the 5 variables they involve are represented by those letters (S = displacement, U = inital velocity, V = final velocity, A = acceleration, T = time)

Linear equations of motion

---

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.

$v\_f\; =\; v\_i\; +\; a\backslash Delta\; t\; \backslash ,$

$d\; =\; \backslash begin\{matrix\}\; \backslash frac\{1\}\{2\}\; \backslash end\{matrix\}\; (v\_i\; +\; v\_f)\backslash Delta\; t$

$d\; =\; v\_i\backslash Delta\; t\; +\; \backslash begin\{matrix\}\; \backslash frac\{1\}\{2\}\; \backslash end\{matrix\}\; a\backslash Delta\; t^2$

$v\_f^2\; =\; v\_i^2\; +\; 2ad\; \backslash ,$

$d\; =\; v\_f\backslash Delta\; t\; -\; \backslash begin\{matrix\}\; \backslash frac\{1\}\{2\}\; \backslash end\{matrix\}\; a\backslash Delta\; t^2$

where...

$v\_i\; \backslash ,$ is the body's initial speed

and its current state is described by:

$d\; \backslash ,$, the distance travelled from initial state

$v\_f\; \backslash ,$, the current speed

$\backslash Delta\; t\; \backslash ,$, the time between the initial and current states

$a$ is 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 remaining two.

Classic version

---

The above equations are often found in the following version:

$v\; =\; u+at\; \backslash ,$

$s\; =\; \backslash frac\; \{1\}\; \{2\}(u+v)\; \backslash cdot\; t$

$s\; =\; ut\; +\; \backslash frac\; \{1\}\; \{2\}\; a\; t^2$

$v^2\; =\; u^2\; +\; 2\; a\; s\; \backslash ,$

$s\; =\; vt\; -\; \backslash frac\; \{1\}\; \{2\}\; a\; t^2$

where

s = the distance travelled from the initial state to the final state (displacement)

u = the initial speed

v = the final speed

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 normal 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:

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

Substituting and cancelling minus signs gives:

$s\; =\; \backslash frac\{u^2\}\{2g\}$

Extension

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

$v\; =\; v\_0\; +\; at\; \backslash ,$

$s\; =\; s\_0\; +\; \backslash begin\{matrix\}\; \backslash frac\{1\}\{2\}\; \backslash end\{matrix\}\; (v\_0\; +\; v)t\; \backslash ,$

$s\; =\; s\_0\; +\; v\_0\; t\; +\; \backslash begin\{matrix\}\; \backslash frac\{1\}\{2\}\; \backslash end\{matrix\}\{at^2\}\; \backslash ,$

$(v)^2\; =\; (v\_0)^2\; +\; 2a\; \backslash Delta\; s\; \backslash ,$

$s\; =\; s\_0\; +\; v\; t\; -\; \backslash begin\{matrix\}\; \backslash frac\{1\}\{2\}\; \backslash end\{matrix\}\{at^2\}\; \backslash ,$

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:

$\backslash omega\; =\; \backslash omega\_0\; +\; \backslash alpha\; t\; \backslash ,$

$\backslash phi\; =\; \backslash phi\_0$

+ \begin{matrix} \frac{1}{2} \end{matrix}(\omega_0 + \omega)t

$\backslash phi\; =\; \backslash phi\_0\; +\; \backslash omega\_0\; t\; +\; \backslash begin\{matrix\}\; \backslash frac\{1\}\{2\}\; \backslash end\{matrix\}\backslash alpha\; \{t^2\}\; \backslash ,$

$(\backslash omega)^2\; =\; (\backslash omega\_0)^2\; +\; 2\backslash alpha\; \backslash Delta\; \backslash phi\; \backslash ,$

$\backslash phi\; =\; \backslash phi\_0\; +\; \backslash omega\; t\; -\; \backslash begin\{matrix\}\; \backslash frac\{1\}\{2\}\; \backslash end\{matrix\}\backslash alpha\; \{t^2\}\; \backslash ,$

where:

$\backslash alpha$ is the angular acceleration

$\backslash omega$ is the angular velocity

$\backslash phi$ is the angular displacement

$\backslash omega\_0$ is the initial angular velocity

$\backslash phi\_0$ is the initial angular displacement

$\backslash Delta\; \backslash phi$ is the variation on angular displacement ($\backslash phi$ - $\backslash phi\_0$).

Derivation

---

Motion equation 1

By definition of acceleration,

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

Hence

$at\; =\; v\; -\; u\; \backslash ,$

$v\; =\; u\; +\; at\; \backslash ,$

Motion equation 2

By definition,

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

Hence

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

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

Motion equation 3

Insert Motion Equation 1 into Motion Equation 2

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

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

$s\; =\; ut\; +\; \backslash begin\{matrix\}\; \backslash frac\{1\}\{2\}\; \backslash end\{matrix\}\; at^2$

Motion equation 4

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

Using Motion Equation 2, replace t with above

$s\; =\; \backslash begin\{matrix\}\; \backslash frac\{1\}\{2\}\; \backslash end\{matrix\}\; (u\; +\; v)\; (\; \backslash frac\{v\; -\; u\}\{a\}\; )$

$2as\; =\; (u\; +\; v)(v\; -\; u)\; \backslash ,$

$2as\; =\; v^2\; -\; u^2\; \backslash ,$

$v^2\; =\; u^2\; +\; 2as\; \backslash ,$

Motion equation 5

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

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

See also

---

• Scalar (physics)
• Vector
• Distance
• Displacement
• Speed
• Velocity
• Acceleration
• Jerk
• Angular displacement
• Angular speed
• Angular velocity
• Angular acceleration
• Constant gravity, no drag or wind
• Newton's laws of motion
• Torricelli's Equation

References

---

• Fundamentals of Physics Robert Resnick, David Halliday, Jearl Walker

Mechanics | Equations

Bewegungsgleichung | Kinematyczne równanie ruchu | Equações de movimento