NationMaster - Encyclopedia: 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. Image File history File links Please see the file description page for further information. ... It has been suggested that this article or section be merged into Equation of motion. ... This article needs additional references or sources for verification. ... This article is about equations in mathematics. ... In physics, force is an influence that may cause an object to accelerate. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ...

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 = initial velocity, V = final velocity, A = acceleration, T = time) Acceleration is the time rate of change of velocity, and at any point on a velocity-time graph, it is given by the slope of the tangent to that point basicly. ... It has been suggested that this article or section be merged into Equation of motion. ...

1 Linear equations of motion
2 Classic version
- 2.1 Examples
- 2.2 Extension
3 Rotational equations of motion
4 Derivation
5 See also
6 References

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.

$v_f = v_i + aDelta t ,$

$d = begin{matrix} frac{1}{2} end{matrix} (v_i + v_f)Delta t$

$d = d_i + v_iDelta t + begin{matrix} frac{1}{2} end{matrix} aDelta t^2$

$v_f^2 = v_i^2 + 2ad ,$

where...

$v_i ,$ is the body's initial speed

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. In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ... Gravity is a force of attraction that acts between bodies that have mass. ... g (also gee, g-force or g-load) is a non-SI unit of acceleration defined as exactly 9. ...

Classic version

The above equations are often found in the following version:

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

$R = frac {1} {2}(u+v) t$ ...(2)

$R = ut + frac {1} {2} a t^2$ ...(3)

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

$R = vt - frac {1} {2} a t^2$ ...(5)

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

where

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

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. A projectile is any object sent through space by the application of a force. ...

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. In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ... Gravity is a force of attraction that acts between bodies that have mass. ...

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

$R= frac{v^2 - u^2}{-2g}$

Substituting and cancelling minus signs gives:

$R = frac{u^2}{2g}$

Extension

More complex versions of these equations can include a quantity $Δ$ s for the variation on displacement (R - R₀), R₀ for the initial position of the body, and v₀ for u for consistency.

$v = v_0 + at ,$

$R = R_0 + begin{matrix} frac{1}{2} end{matrix} (v_0 + v)t ,$

$R = R_0 + v_0 t + begin{matrix} frac{1}{2} end{matrix}{at^2} ,$

$(v)^2 = (v_0)^2 + 2a Delta R ,$

$R = R_0 + v t - begin{matrix} frac{1}{2} end{matrix}{at^2} ,$

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: A sphere rotating around its axis. ...

$omega = omega_0 + alpha t ,$

$phi = phi_0 + begin{matrix} frac{1}{2} end{matrix}(omega_0 + omega)t$

$phi = phi_0 + omega_0 t + begin{matrix} frac{1}{2} end{matrix}alpha {t^2} ,$

$(omega)^2 = (omega_0)^2 + 2alpha Delta phi ,$

$phi = phi_0 + omega t - begin{matrix} frac{1}{2} end{matrix}alpha {t^2} ,$

where:

α

is the angular acceleration

ω

is the angular velocity

φ

is the angular displacement

ω 0

is the initial angular velocity

φ 0

is the initial angular displacement

Δφ

is the variation on angular displacement (

φ

φ 0

Ð»Insert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ... Rotation of a rigid object P about a fixed object about a fixed axis O. Angular displacement of a body is the angle in radians (degrees, revolutions) through which a point or line has been rotated in a specified sense about a specified axis. ...

Derivation

Motion equation 1

By definition of acceleration,

$a = frac{Delta v}{Delta t}quadRightarrowquad a = frac{v - u}{t}$

Hence

$at = v - u ,$

$v = u + at ,$

Motion equation 2

By definition,

$mathrm{ average velocity } = frac{R}{t}$

Hence

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

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

Motion equation 3

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

Using Motion Equation 2, replace t with above

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

Motion equation 4

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

In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ... This article does not cite any references or sources. ... In physics, velocity is defined as the rate of change of displacement or the rate of displacement. ... Acceleration is the time rate of change of velocity, and at any point on a velocity-time graph, it is given by the slope of the tangent to that point basicly. ... It has been suggested that this article or section be merged into Equation of motion. ... Look up jerk, jolt, surge, lurch in Wiktionary, the free dictionary. ... Rotation of a rigid object P about a fixed object about a fixed axis O. Angular displacement of a body is the angle in radians (degrees, revolutions) through which a point or line has been rotated in a specified sense about a specified axis. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ... Ð»Insert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non... Mathematically the term trajectory refers to the ordered set of states which are assumed by a dynamical system over time (see e. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... This equation was created by Evangelista Torricelli to find the final velocity of a moving object without having a known time interval and was named after him. ...

References

Halliday, David, Robert Resnick and Jearl Walker, Fundamentals of Physics, Wiley; 7 Sub edition (June 16, 2004). ISBN 0471232319.

Categories: Articles to be merged since March 2007 | Classical mechanics | Equations

Results from FactBites:

Equation of motion - Wikipedia, the free encyclopedia (503 words)
	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.

Equations of Motion (3750 words)
	The equations of motion are valid only when acceleration is constant and motion is constrained to a straight line.
	Motion along a curved path may also be effectively one-dimensional if there is only one degree of freedom for the objects involved.
	The second equation of motion is a quadratic and solving it for time would introduce a lot of nastiness into the algebra.

More results at FactBites »

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