Equations of motion

Classical mechanics
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Core topics Rigid body · Rigid body dynamics · Euler's equations (rigid body dynamics) · Motion · Newton's laws of motion · Newton's law of universal gravitation · Euler's laws of motion · Equations of motion · Inertial frame of reference · Non-inertial reference frame · Rotating reference frame · Fictitious force · Linear motion · Mechanics of planar particle motion · Displacement (vector) · Relative velocity · Friction · Simple harmonic motion · Harmonic oscillator · Vibration · Damping · Damping ratio · Rotational motion · Circular motion · Uniform circular motion · Non-uniform circular motion · Centripetal force · Centrifugal force · Centrifugal force (rotating reference frame) · Reactive centrifugal force · Coriolis force · Pendulum · Rotational speed · Angular acceleration · Angular velocity · Angular frequency · Angular displacement
Scientists Galileo Galilei · Isaac Newton · Jeremiah Horrocks · Leonhard Euler · Jean le Rond d'Alembert · Alexis Clairaut · Joseph Louis Lagrange · Pierre-Simon Laplace · William Rowan Hamilton · Siméon-Denis Poisson
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Equations of motion are equations that describe the behavior of a system in terms of its motion as a function of time (e.g., the motion of a particle under the influence of a force).^[1] 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.

Equations of uniformly accelerated linear motion

The equations that apply to bodies moving linearly (in one dimension) with constant acceleration are often referred to as "SUVAT" equations where the five variables are represented by those letters (s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time); the five letters may be shown in a different order.

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

$v = v_i + a \Delta t \,$ $v^2 = v_i^2 + 2a(s - s_i) \,$ $s = s_i + v_i\Delta t + \tfrac{1}{2} a(\Delta t)^2 \,$ $s = s_i + v\Delta t - \tfrac{1}{2} a(\Delta t)^2 \,$ $\Delta t = \tfrac{2s - 2s_i}{v + v_i} \,$

where... $v_i \,$ is the body's initial velocity $s_i \,$ is the body's initial position and its current state is described by: $v \,$, The velocity at the end of the interval $s \,$, the position at the end of the interval (displacement) $\Delta t \,$, the time interval 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. Thus, in this situation it is sufficient to know three out of the five variables to calculate the remaining two.

Classic version

The equations above are often written in the following form^[2] (often informally known as the "suvat"^[3] equations):

$\begin{alignat}{3} & v && = u+at \qquad & \text{(1)} \\ & s && = \tfrac12(u+v)t \qquad & \text{(2)} \\ & s && = ut + \tfrac12 at^2 \qquad & \text{(3)} \\ & s && = vt - \tfrac12 at^2 \qquad & \text{(4)} \\ & v^2 && = u^2 + 2as \qquad & \text{(5)} \\ & a && = \frac{v-u}{t} \qquad & \text{(6)} \\ \end{alignat}$

By substituting (1) into (2), we can get (3), (4) and (5). (6) can be constructed by rearranging (1).

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 fifth 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 s₀ for the initial position of the body, and v₀ instead of u for consistency.

$v = v_0 + at \,$

$s = s_0 + \tfrac{1}{2} (v_0 + v)t \,$

$s = s_0 + v_0 t + \tfrac{1}{2} at^2 \,$

$v^2 = v_0^2 + 2a(s - s_0) \,$

$s = s_0 + vt - \tfrac{1}{2} at^2 \,$

Equations of circular motion

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

$\omega = \omega_0 + \alpha t\,$

$\phi = \phi_0 + \tfrac12(\omega_0 + \omega)t$

$\phi = \phi_0 + \omega_0t + \tfrac12\alpha t^2\,$

$\omega^2 = \omega_0^2 + 2\alpha(\phi - \phi_0)\,$

$\phi = \phi_0 + \omega t - \tfrac12\alpha t^2\,$

where:

$\alpha\,$ is the angular acceleration

$\omega\,$ is the angular velocity

$\phi\,$ is the angular displacement

$\omega_0\,$ is the initial angular velocity.

t is the time taken to move from the initial state to the final state

Derivation

These equations assume constant acceleration and non-relativistic velocities.

Equation 2

By definition:

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

Hence:

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

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

Equation 4

Using equation 1 to substitute u in equation 2 gives:

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

Equation 5

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

Using equation 2, substitute t with above:

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

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

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

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

See also

• Scalar (physics)
• Vector
• Distance
• Displacement
• Speed
• Velocity
• Acceleration
• Angular displacement
• Angular speed
• Angular velocity
• Angular acceleration
• Equations for a falling body
• Parabolic trajectory
• Newton's laws of motion
• Torricelli's Equation
• Euler–Lagrange equation

External links

• Equations of Motion Applet

References

1. ^ Halliday, David; Resnick, Robert; Walker, Jearl (2004-06-16). Fundamentals of Physics (7 Sub ed.). Wiley. ISBN 0471232319. 
2. ^ Hanrahan, Val; Porkess, R (2003). Additional Mathematics for OCR. London: Hodder & Stoughton. p. 219. ISBN 0-340-86960-7. 
3. ^ Keith Johnson (2001). Physics for you: revised national curriculum edition for GCSE (4th ed.). Nelson Thornes. p. 135. ISBN 9780748762361. http://books.google.com/books?id=D4nrQDzq1jkC&pg=PA135&dq=suvat#v=onepage&q=suvat&f=false. "You can remember the 5 symbols by 'suvat'. If you know any three of 'suvat', the other two can be found."