A 'g' is a measure of acceleration, the rate of change of velocity. So you could be either rapidly stopping, speeding up, or turning to generate large G force ratings.
In physics, acceleration
is the rate at which the velocity of a body changes with time. In general, velocity and acceleration are vector quantities, with magnitude and direction, though in many cases only magnitude is considered (sometimes with negative values for deceleration, treating it as a one dimensional vector). As described by Newton's Second Law, acceleration is caused by a net force; the force, as a vector, is equal to the product of the mass of the object being accelerated (scalar) and the acceleration (vector). The SI unit of acceleration is the meter per second squared (m/s2).
For example, an object such as a car that starts from standstill, then travels in a straight line at increasing speed, is accelerating in the direction of travel. If the car changes direction at constant speedometer reading, there is strictly speaking an acceleration although it is often not so described; passengers in the car will experience a force pushing them back into their seats in linear acceleration, and a sideways force on changing direction. If the speed of the car decreases, it is usual and meaningful to speak of deceleration
; mathematically it is acceleration in the opposite direction to that of motion.
Mathematically, instantaneous acceleration—acceleration over an infinitesimal interval of time—is the rate of change of velocity over time:
(Here and elsewhere, if motion is in a straight line, vector quantities can be substituted by scalars in the equations.)
Average acceleration over a period of time is the change in velocity
divided by the duration of the period
Acceleration has the dimensions of velocity (L/T) divided by time, i.e., L/T2. The SI unit of acceleration is the metre per second squared (m/s2); this can be called more meaningfully "metre per second per second", as the velocity in metres per second changes by the acceleration value, every second.
An object moving in a circular motion—such as a satellite orbiting the earth—is accelerating due to the change of direction of motion, although the magnitude (speed) may be constant. When an object is executing such a motion where it changes direction, but not speed, it is said to be undergoing centripetal
(directed towards the center) acceleration. Oppositely, a change in the speed of an object, but not its direction of motion, is a tangential
Proper acceleration, the acceleration of a body relative to a free-fall condition, is measured by an instrument called an accelerometer.
In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net force vector (i.e., sum of all forces) acting on it (Newton's second law):
is the net force acting on the body, m
is the mass of the body, and a
is the center-of-mass acceleration. As speeds approach the speed of light, relativistic effects become increasingly large and acceleration becomes less.
The velocity of a particle moving on a curved path as a function of time can be written as:
) equal to the speed of travel along the path, and
a unit vector tangent to the path pointing in the direction of motion at the chosen moment in time. Taking into account both the changing speed v(t)
and the changing direction of ut
, the acceleration of a particle moving on a curved path can be written using the chain rule of differentiation for the product of two functions of time as:
is the unit (inward) normal vector to the particle's trajectory (also called the principal normal
), and r
is its instantaneous radius of curvature based upon the osculating circle at time t
. These components are called the tangential acceleration and the normal or radial acceleration (or centripetal acceleration in circular motion, see also circular motion and centripetal force).
Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the Frenet–Serret formulas.
acceleration is a type of motion in which the velocity of an object changes by an equal amount in every equal time period.
A frequently cited example of uniform acceleration is that of an object in free fall in a uniform gravitational field. The acceleration of a falling body in the absence of resistances to motion is dependent only on the gravitational field strength g
(also called acceleration due to gravity
). By Newton's Second Law the force, F
, acting on a body is given by:
Due to the simple algebraic properties of constant acceleration in the one-dimensional case (that is, the case of acceleration aligned with the initial velocity), there are simple formulas that relate the following quantities: displacement, initial velocity u
, final velocity v
, acceleration a
, and time t
In the case of uniform acceleration of an object that is initially moving in a direction not aligned with the acceleration, the motion can be resolved into two orthogonal parts, one of constant velocity and the other according to the above equations. As Galileo showed, the net result is parabolic motion, as in the trajectory of a cannonball, neglecting air resistance.
Uniform circular motion, that is constant speed along a circular path, is an example of a body experiencing acceleration resulting in velocity of a constant magnitude but change of direction. In this case, because the direction of the object's motion is constantly changing, being tangential to the circle, the object's linear velocity vector also changes, but its speed does not. This acceleration is a radial acceleration since it is always directed toward the centre of the circle and takes the magnitude:
is the object's linear speed along the circular path. Equivalently, the radial acceleration vector (
) may be calculated from the object's angular velocity
is a vector directed from the centre of the circle and equal in magnitude to the radius. The negative shows that the acceleration vector is directed towards the centre of the circle (opposite to the radius).
The acceleration, hence also the net force acting on a body in uniform circular motion, is directed toward
the centre of the circle; that is, it is centripetal. Whereas the so-called 'centrifugal force' appearing to act outward on the body is really a pseudo force experienced in the frame of reference of the body in circular motion, due to the body's linear momentum at a tangent to the circle.
With nonuniform circular motion, i.e., the speed along the curved path changes, a transverse accleration is produced equal to the rate of change of the angular speed around the circle times the radius of the circle. That is,
The transverse (or tangential) acceleration is directed at right angles to the radius vector and takes the sign of the angular acceleration (
The special theory of relativity describes the behavior of objects traveling relative to other objects at speeds approaching that of light in a vacuum. Newtonian mechanics is exactly revealed to be an approximation to reality, valid to great accuracy at lower speeds. As the relevant speeds increase toward the speed of light, acceleration no longer follows classical equations.
As speeds approach that of light, the acceleration produced by a given force decreases, becoming infinitesimally small as light speed is approached; an object with mass can approach this speed asymptotically, but never reach it.
Unless the state of motion of an object is known, it is totally impossible to distinguish whether an observed force is due to gravity or to acceleration—gravity and inertial acceleration have identical effects. Albert Einstein called this the principle of equivalence, and said that only observers who feel no force at all—including the force of gravity—are justified in concluding that they are not accelerating.
In physics, the angular velocity
is defined as the rate of change of angular displacement and is a vector quantity (more precisely, a pseudovector) which specifies the angular speed (rotational speed) of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, degrees per hour, etc. Angular velocity is usually represented by the symbol omega (ω
, rarely Ω
The direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction which is usually specified by the right-hand rule.
The angular velocity of a particle is measured around or relative to a point, called the origin. As shown in the diagram (with angles ɸ
in radians), if a line is drawn from the origin (O) to the particle (P), then the velocity (v
) of the particle has a component along the radius (radial component, v‖
) and a component perpendicular to the radius (cross-radial component, v⊥
). If there is no radial component, then the particle moves in a circle. On the other hand, if there is no cross-radial component, then the particle moves along a straight line from the origin.
A radial motion produces no change in the direction of the particle relative to the origin, so for purposes of finding the angular velocity the radial component can be ignored. Therefore, the rotation is completely produced by the perpendicular motion around the origin, and the angular velocity is completely determined by this component.
In two dimensions the angular velocity ω
is given by
This is related to the cross-radial (tangential) velocity by:
An explicit formula for v⊥
in terms of v
Combining the above equations gives a formula for ω
In two dimensions the angular velocity is a single number which has no direction, but it does have a sense or orientation. In two dimensions the angular velocity is a pseudoscalar, a quantity which changes its sign under a parity inversion (for example if one of the axes is inverted or they are swapped). The positive direction of rotation is taken, by convention, to be in the direction towards the y
axis from the x
axis. If parity is inverted, but the sense of a rotation does not, then the sign of the angular velocity changes.
There are three types of angular velocity involved in the movement on an ellipse corresponding to the three anomalies (true, eccentric and mean).
In three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in this case is generally thought of as a vector, or more precisely, a pseudovector. It now has not only a magnitude, but a direction as well. The magnitude is the angular speed, and the direction describes the axis of rotation. The right-hand rule indicates the positive direction of the angular velocity pseudovector.
an unitary vector over the instantaneous rotation axis, so that from the top of the vector the rotation is counter-clock-wise the angular velocity vector
can be defined as:
Just as in the two dimensional case, a particle will have a component of its velocity along the radius from the origin to the particle, and another component perpendicular to that radius. The combination of the origin point and the perpendicular component of the velocity defines a plane of rotation in which the behavior of the particle (for that instant) appears just as it does in the two dimensional case. The axis of rotation is then a line normal to this plane, and this axis defined the direction of the angular velocity pseudovector, while the magnitude is the same as the pseudoscalar value found in the 2-dimensional case. Using the unit vector
defined before, the angular velocity vector may be written in a manner similar to that for two dimensions:
which, by the definition of the cross product, can be written:
If a point rotates with
in a frame
which rotates itself with angular speed
respect an external frame
, we can define the addition of
like the angular velocity vector of the point respect
With this operation defined like this, angular velocity, which is a pseudovector, becomes also a real vector because it has two operations:
This is the definition of a vector space. The only property that presents difficulties to prove is the commutativity of the addition. This can be proven from the fact that the velocity tensor W (see below) is skew-symmetric. Therefore
is a rotation matrix and in a time dt is an infinitesimal rotation matrix. Therefore it can be expanded as
The composition of rotations is not commutative, but when they are infinitesimal rotations the first order approximation of the previous series can be taken and
, and therefore
Given a rotating frame composed by three unitary vectors, all the three must have the same angular speed in any instant. In such a frame each vector is a particular case of the previous case (moving particle), in which the module of the vector is constant.
Though it is just a particular case of the previous one, is a very important one for its relationship with the rigid body study, and special tools have been developed for this case. There are two possible ways to describe the angular velocity of a rotating frame. The angular velocity vector and the angular velocity tensor. Both entities are related and they can be calculated from each other.
It is defined as the angular velocity of each of the vectors of the frame, in a consistent way with the general definition.
It is known by the Euler's rotation theorem that for a rotating frame there exists an instantaneous axis of rotation in any instant. In the case of a frame, the angular velocity vector is over the instantaneous axis of rotation.
Any transversal section of a plane perpendicular to this axis has to behave as a two dimensional rotation. Thus, the magnitude of the angular velocity vector at a given time t
is consistent with the two dimensions case.
Angular velocity is a vector defining an addition operation. Components can be calculated from the derivatives of the parameters defining the moving frame (Euler angles or rotation matrices)
As in the general case, the addition operation for angular velocity vectors can be defined using movement composition. In the case of rotating frames, the movement composition is simpler than the general case because the final matrix is always a product of rotation matrices.
As in the general case, addition is commutative
Substituting in the expression
any vector e of the frame we obtain
, and therefore
As the columns of the matrix of the frame are the components of its vectors, this allows also to calculate
from the matrix of the frame and its derivative.
The components of the angular velocity pseudovector were first calculated by Leonhard Euler using his Euler angles and an intermediate frame made out of the intermediate frames of the construction:
Euler proved that the projections of the angular velocity pseudovector over these three axes was the derivative of its associated angle (which is equivalent to decompose the instant rotation in three instantaneous Euler rotations). Therefore:
This basis is not orthonormal and it is difficult to use, but now the velocity vector can be changed to the fixed frame or to the moving frame with just a change of bases. For example, changing to the mobile frame:
where IJK are unit vectors for the frame fixed in the moving body. This example has been made using the Z-X-Z convention for Euler angles.
The components of the angular velocity vector can be calculated from infinitesimal rotations (if available) as follows:
It can be introduced from rotation matrices. Any vector
that rotates around an axis with an angular speed vector
(as defined before) satisfies:
We can introduce here the angular velocity tensor
associated to the angular speed
This tensor W(t) will act as if it were a
Given the orientation matrix A(t) of a frame, we can obtain its instant angular velocity tensor
W as follows. We know that:
As angular speed must be the same for the three vectors of a rotating frame, if we have a matrix A(t) whose columns are the vectors of the frame, we can write for the three vectors as a whole:
And therefore the angular velocity tensor we are looking for is:
In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor which is a second rank skew-symmetric tensor.
This tensor W will have n(n-1)/2 independent components and this number is the dimension of the Lie algebra of the Lie group of rotations of an n
-dimensional inner product space.
In three dimensions angular velocity can be represented by a pseudovector because second rank tensors are dual to pseudovectors in three dimensions.
. This can be read as a differential equation that defines A(t) knowing W(t).
And if the angular speed is constant then is also constant and the equation can be integrated. The result is:
which shows a connection with the Lie group of rotations.
It is possible to prove that angular velocity tensor are skew symmetric matrices which means that a
To prove it we start taking the time derivative of
being R(t) a rotation matrix:
Applying the formula (AB
)t = B
is the negative of its transpose, which implies it is a skew symmetric matrix.
The tensor is a matrix with this structure:
As it is a skew symmetric matrix it has a Hodge dual vector which is precisely the previous angular velocity vector
At any instant,
, the angular velocity tensor represents a linear map between the position vectors
and their velocity vectors
of a rigid body rotating around the origin:
where we omitted the
parameter, and regard
as elements of the same 3-dimensional Euclidean vector space
The relation between this linear map and the angular velocity pseudovector
is the following.
Because of W
is the derivative of an orthogonal transformation, the
bilinear form is skew-symmetric. (Here
stands for the scalar product). So we can apply the fact of exterior algebra that there is a unique linear form
is the wedge product of
Taking the dual vector L
* of L
, as the Hodge dual of L
*, and apply further Hodge dual identities we arrive at
is an arbitrary vector, from nondegeneracy of scalar product follows
For angular velocity tensor maps velocities to positions, it is a vector field. In particular, this vector field is a Killing vector field belonging to an element of the Lie algebra so(3) of the 3-dimensional rotation group SO(3). This element of so(3) can also be regarded as the angular velocity vector.
The same equations for the angular speed can be obtained reasoning over a rotating rigid body. Here is not assumed that the rigid body rotates around the origin. Instead it can be supposed rotating around an arbitrary point which is moving with a linear velocity V(t) in each instant.
To obtain the equations it is convenient to imagine a rigid body attached to the frames and consider a coordinate system that is fixed with respect to the rigid body. Then we will study the coordinate transformations between this coordinate and the fixed "laboratory" system.
As shown in the figure on the right, the lab system's origin is at point O, the rigid body system origin is at O' and the vector from O to O' is R
. A particle (i
) in the rigid body is located at point P and the vector position of this particle is Ri
in the lab frame, and at position ri
in the body frame. It is seen that the position of the particle can be written:
The defining characteristic of a rigid body is that the distance between any two points in a rigid body is unchanging in time. This means that the length of the vector
is unchanging. By Euler's rotation theorem, we may replace the vector
is a 3x3 rotation matrix and
is the position of the particle at some fixed point in time, say t=0. This replacement is useful, because now it is only the rotation matrix
which is changing in time and not the reference vector
, as the rigid body rotates about point O'. Also, since the three columns of the rotation matrix represent the three versors of a reference frame rotating together with the rigid body, any rotation about any axis becomes now visible, while the vector
would not rotate if the rotation axis were parallel to it, and hence it would only describe a rotation about an axis perpendicular to it (i.e., it would not see the component of the angular velocity pseudovector parallel to it, and would only allow the computation of the component perpendicular to it). The position of the particle is now written as:
Taking the time derivative yields the velocity of the particle:
is the velocity of the particle (in the lab frame) and V
is the velocity of O' (the origin of the rigid body frame). Since
is a rotation matrix its inverse is its transpose. So we substitute
is the previous angular velocity tensor.
It can be proved that this is skew symmetric matrix, so we can take its dual to get a 3 dimensional pseudovector which is precisely the previous angular velocity vector
Substituting ω for W
into the above velocity expression, and replacing matrix multiplication by an equivalent cross product:
It can be seen that the velocity of a point in a rigid body can be divided into two terms – the velocity of a reference point fixed in the rigid body plus the cross product term involving the angular velocity of the particle with respect to the reference point. This angular velocity is the "spin" angular velocity of the rigid body as opposed to the angular velocity of the reference point O' about the origin O.
We have supposed that the rigid body rotates around an arbitrary point. We should prove that the angular velocity previously defined is independent from the choice of origin, which means that the angular velocity is an intrinsic property of the spinning rigid body.
See the graph to the right: The origin of lab frame is O, while O1
are two fixed points on the rigid body, whose velocity is
respectively. Suppose the angular velocity with respect to O1
respectively. Since point P and O2
have only one velocity,
The above two yields that
Since the point P (and thus
) is arbitrary, it follows that
If the reference point is the instantaneous axis of rotation the expression of velocity of a point in the rigid body will have just the angular velocity term. This is because the velocity of instantaneous axis of rotation is zero. An example of instantaneous axis of rotation is the hinge of a door. Another example is the point of contact of a pure rolling spherical rigid body.
The terminal velocity
of a falling object is the velocity of the object when the sum of the drag force (Fd
) and buoyancy equals the downward force of gravity (FG
) acting on the object. Since the net force on the object is zero, the object has zero acceleration.
In fluid dynamics, an object is moving at its terminal velocity
if its speed is constant due to the restraining force exerted by the fluid through which it is moving. (The term "terminal velocity" is used rather than "terminal speed" even though we are not concerned with a vector, but just a scalar value.)
As the speed of an object increases, the drag force acting on the object, resultant of the substance (e.g., air or water) it is passing through, increases. At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). At this point the object ceases to accelerate and continues falling at a constant speed called terminal velocity (also called settling velocity). An object moving downward with greater than terminal velocity (for example because it was thrown downwards or it fell from a thinner part of the atmosphere or it changed shape) will slow down until it reaches terminal velocity. Drag depends on the projected area, and this is why objects with a large projected area relative to mass, such as parachutes, have a lower terminal velocity than objects with a small projected area relative to mass, such as bullets.
Based on wind resistance, for example, the terminal velocity of a skydiver in a belly-to-earth (i.e., face down) free-fall position is about 195 km/h (122 mph or 54 m/s). This velocity is the asymptotic limiting value of the acceleration process, because the effective forces on the body balance each other more and more closely as the terminal velocity is approached. In this example, a speed of 50% of terminal velocity is reached after only about 3 seconds, while it takes 8 seconds to reach 90%, 15 seconds to reach 99% and so on.
Higher speeds can be attained if the skydiver pulls in his or her limbs (see also freeflying). In this case, the terminal velocity increases to about 320 km/h (200 mph or 90 m/s), which is almost the terminal velocity of the Peregrine Falcon diving down on its prey. The same terminal velocity is reached for a typical .30-06 bullet dropping downwards—when it is returning to earth having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study.
Competition speed skydivers fly in the head down position and reach even higher speeds. The current world record is 1,357.6km/h (843.6mph/Mach 1.25) by Felix Baumgartner who skydived from 38,969.4m (127,852.4ft) above earth on 14 October 2012. The record was set due to the high altitude where the lesser density of the atmosphere decreased drag.
Mathematically, terminal velocity—without considering buoyancy effects—is given by
Mathematically, an object approaches its terminal velocity asymptotically.
Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes' principle: the mass
has to be reduced by the displaced fluid mass
the volume of the object. So instead of
use the reduced mass
in this and subsequent formulas.
On Earth, the terminal velocity of an object changes due to the properties of the fluid, the mass of the object and its projected cross-sectional surface area.
Air density increases with decreasing altitude, ca. 1% per 80 metres (260 ft) (see barometric formula). For objects falling through the atmosphere, for every 160 metres (520 ft) of falling, the terminal velocity decreases 1%. After reaching the local terminal velocity, while continuing the fall, speed decreases
to change with the local terminal velocity.
Mathematically, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation):
At equilibrium, the net force is zero (F = 0);
Solving for v
The drag equation is
A more practical form of this equation can be obtained by making the substitution k
Dividing both sides by m
The equation can be re-arranged into
Taking the integral of both sides yields
= ( ).
After integration, this becomes
or in simpler a form
The inverse hyperbolic tangent is defined as:
So the solution of the integral is
with tanh the hyperbolic tangent function. Assuming that g
is positive (which it was defined to be), and substituting α
back in, the velocity v
Next, after k
has been substituted, the velocity v
is in the desired form:
As time tends to infinity ( t
→ ∞ ), the hyperbolic tangent tends to 1, resulting in the terminal velocity
For very slow motion of the fluid, the inertia forces of the fluid are negligible (assumption of massless fluid) in comparison to other forces. Such flows are called creeping flows and the condition to be satisfied for the flows to be creeping flows is the Reynolds number,
. The equation of motion for creeping flow (simplified Navier-Stokes equation) is given by
The analytical solution for the creeping flow around a sphere was first given by Stokes in 1851. From Stokes' solution, the drag force acting on the sphere can be obtained as
where the Reynolds number,
. The expression for the drag force given by equation (6) is called Stokes' law.
When the value of
is substituted in the equation (5), we obtain the expression for terminal velocity of a spherical object moving under creeping flow conditions:
The creeping flow results can be applied in order to study the settling of sediment particles near the ocean bottom and the fall of moisture drops in the atmosphere. The principle is also applied in the falling sphere viscometer, an experimental device used to measure the viscosity of high viscous fluids.
In principle one doesn't know beforehand whether to apply the creeping flow solution, or what coefficient of drag to use, because the coefficient depends on the speed. What one can do in this situation is to calculate the product of the coefficient of drag and the square of the Reynolds number:
where ν is the kinematic viscosity, equal to μ/ρ. This product is a function of Reynolds number, and one can consult a graph of Cd
versus Re to find where along the curve the product attains the correct value (a qualitative example of such a graph for spheres is found at this NASA site: ) From this one knows the coefficient of drag and one can then use the formula given higher up to find the terminal velocity.
For a spherical object, the above-mentioned product can be simplified:
We can see from this that the regime and the drag coefficient depend only on the sphere's weight and the fluid properties. There are three regimes: creeping flow, intermediate-Reynolds number Newton's Law (almost constant drag coefficient), and a high-Reynolds number regime. In the latter regime the boundary layer is everywhere turbulent (see Golf ball#Aerodynamics). These regimes are given in the following table. The weight range for each regime is given for water and air at 1 atm pressure and 25 °C. Note though that the weight (given in terms of mass in standard gravity) is the weight in the fluid, which is less than the mass times the local gravity because of buoyancy.
Between the first two regimes there is a smooth transition. But notice that there is overlap between the ranges of Cd
Re2 for the last two regimes. Spheres in this weight range have two stable terminal velocities. A rough surface, such as of a dimpled golf ball, allows transition to the lower drag coefficient at a lower Reynolds number.
When the buoyancy effects are taken into account, an object falling through a fluid under its own weight can reach a terminal velocity (settling velocity) if the net force acting on the object becomes zero. When the terminal velocity is reached the weight of the object is exactly balanced by the upward buoyancy force and drag force. That is
If the falling object is spherical in shape, the expression for the three forces are given below:
Substitution of equations (2–4) in equation (1) and solving for terminal velocity,
to yield the following expression
The metre (or meter) per second squared is the unit of acceleration in the International System of Units (SI). As a derived unit it is composed from the SI base units of length, the metre, and the standard unit of time, the second. Its symbol is written in several forms as m/s2, m·s−2, or m s−2.
As acceleration, the unit is interpreted physically as change in velocity or speed per time interval, i.e. metre per second per second and is treated as a vector quantity.
An object experiences a constant acceleration of one metre per second squared (1 m/s2) from a state of rest, when it achieves the speed of 5 m/s after 5 seconds and 10 m/s after 10 seconds.
Newton's Second Law states that force equals mass multiplied by acceleration. The unit of force is the newton (N), and mass has the SI unit kilogram (kg). One newton equals one kilogram metre per second squared. Therefore, the unit metre per second squared is equivalent to newton per kilogram, N·kg−1, or N/kg.
Thus, the Earth's gravitational field (near ground level) can be quoted as 9.8 metres per second squared, or the equivalent 9.8 N/kg.
Acceleration can be measured in ratios to gravity, such as g-force, and peak ground acceleration in earthquakes.
In kinematics, velocity
is the rate of change of the position of an object, equivalent to a specification of its speed and direction of motion. For motion in one dimension, velocity can be defined as the slope of the position vs. time graph of an object. Speed describes only how fast an object is moving, whereas velocity gives both how fast and in what direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified. To have a constant velocity
, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path (the object's path does not curve). Thus, a constant velocity means motion in a straight line at a constant speed. If there is a change in speed, direction, or both, then the object is said to have a changing velocity and is undergoing an acceleration
. For example, a car moving at a constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration.
Velocity is a vector physical quantity; both magnitude and direction are required to define it. The scalar absolute value (magnitude) of velocity is called "speed", a quantity that is measured in metres per second (m/s or m⋅s−1) when using the SI (metric) system. For example, "5 metres per second" is a scalar (not a vector), whereas "5 metres per second east" is a vector. The rate of change of velocity (in m/s) as a function of time (in s) is "acceleration" (in m/s2 – stated "metres per second per second"), which describes how an object's speed and direction of travel change at each point in time.
of an object moving through a displacement
during a time interval
is described by the formula:
The velocity vector v
of an object that has positions x
) at time t
, can be computed as the derivative of position:
Velocity is also defined as rate of change of displacement. Average velocity magnitudes are always smaller than or equal to average speed of a given particle. Instantaneous velocity is always tangential to trajectory. Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.
The equation for an object's velocity can be obtained mathematically by evaluating the integral of the equation for its acceleration beginning from some initial period time
to some point in time later
The final velocity v
of an object which starts with velocity u
and then accelerates at constant acceleration a
for a period of time
The average velocity of an object undergoing constant acceleration is
, where u
is the initial velocity and v
is the final velocity. To find the position, x
, of such an accelerating object during a time interval,
When only the object's initial velocity is known, the expression,
can be used.
This can be expanded to give the position at any time t
in the following way:
These basic equations for final velocity and position can be combined to form an equation that is independent of time, also known as Torricelli's equation:
The above equations are valid for both Newtonian mechanics and special relativity. Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on the value of t
and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words only relative velocity can be calculated.
In Newtonian mechanics, the kinetic energy (energy of motion),
, of a moving object is linear with both its mass and the square of its velocity:
The kinetic energy is a scalar quantity.
is the minimum a ballistic object needs to escape from a massive body like the earth. It represents the kinetic energy that when added to the object's gravitational potential energy (which is always negative) is greater than or equal to zero. Escape velocity from the Earth's surface is about 11 100 m/s.
is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame.
If an object A is moving with velocity vector v
and an object B with velocity vector w
, then the velocity of object A relative to
object B is defined as the difference of the two velocity vectors:
Similarly the relative velocity of object B moving with velocity w
, relative to object A moving with velocity v
Usually the inertial frame is chosen in which the latter of the two mentioned objects is in rest.
In the one dimensional case, the velocities are scalars and the equation is either:
In polar coordinates, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin (also known as velocity made good
), and an angular velocity, which is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system).
The radial and angular velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The transverse velocity is the component of velocity along a circle centered at the origin.
The magnitude of the radial velocity
is the dot product of the velocity vector and the unit vector in the direction of the displacement.
The magnitude of the transverse velocity
is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. It is also the product of the angular speed
and the magnitude of the displacement.
Angular momentum in scalar form is the mass times the distance to the origin times the transverse velocity, or equivalently, the mass times the distance squared times the angular speed. The sign convention for angular momentum is the same as that for angular velocity.
is known as moment of inertia. If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion.
The drift velocity
is the average velocity that a particle, such as an electron, attains due to an electric field. It can also be referred to as axial drift velocity since particles defined are assumed to be moving along a plane. In general, an electron will ‘rattle around’ in a conductor at the Fermi velocity randomly. An applied electric field will give this random motion a small net velocity in one direction.
In a semiconductor, the two main carrier scattering mechanisms are ionized impurity scattering and lattice scattering.
Because current is proportional to drift velocity, which is, in turn, proportional to the magnitude of an external electric field, Ohm's law can be explained in terms of drift velocity.
Drift velocity is expressed in the following equations:
is the current density,
is charge density (in units C/m3), and
is the drift velocity, and where
is the electron mobility (in units (m^2)/V*s) and
is the electric field (in units V/m).
The formula for evaluating the drift velocity of charge carriers in a material of constant cross-sectional area is given by]
where is the drift velocity of electrons, is the current flowing through the material, is the charge-carrier density, is the area of cross-section of the material and is the charge on the charge-carrier.
In terms of the basic properties of the right-cylindrical current-carrying metallic conductor, where the charge-carriers are electrons, this expression can be rewritten as]
Electricity is most commonly conducted in a copper wire. Copper has a density of 8.94 g/cm³, and an atomic weight of 63.546 g/mol, so there are 140685.5 mol/m³. In 1 mole of any element there are 6.02×1023 atoms (Avogadro's constant). Therefore in 1m³ of copper there are about 8.5×1028 atoms (6.02×1023 × 140685.5 mol/m³). Copper has one free electron per atom, so n
is equal to 8.5×1028 electrons per m³.
Assume a current I
= 3 amperes, and a wire of 1 mm diameter (radius in meters = 0.0005m). This wire has a cross sectional area of 7.85×10−7 m2 (A
= π×0.00052). The charge of 1 electron is q
=−1.6×10−19 Coulombs. The drift velocity therefore can be calculated:
[v] = [amperes] / ( [electron/m3] × [m2] × [coulombs/electron] )
Therefore in this wire the electrons are flowing at the rate of −0.00029 m/s, or very nearly −1.0 m/hour.
By comparison, the Fermi velocity of these electrons (which, at room temperature, can be thought of as their approximate velocity in the absence of electric current) is around 1570 km/s.
In the case of alternating current, the direction of electron drift switches with the frequency of the current. In the example above, if the current were to alternate with the frequency of F = 60 Hz, drift velocity would likewise vary in a sine-wave pattern, and electrons would fluctuate about their initial positions with the amplitude of:
Rotational speed (sometimes called speed of revolution) is the number of complete rotations or revolutions per time unit. Rotational speed is a cyclic frequency measured in hertz ("rotations per minute or per second") in the SI System or revolutions per minute or per second (rpm or 1/min), the latter of which is more common in everyday life. Scientists, however, generally prefer to measure rotational speed in radians per second. The symbol for rotational speed is ω (the Greek lowercase letter "omega").
When proper units are used for tangential speed v, rotational speed ω, and radial distance r, the direct proportion of v to both r and ω becomes the exact equation:
An algebraic rearrangement of this equation allows us to solve for rotational speed:
Thus, the tangential speed will be directly proportional to r when all parts of a system simultaneously have the same ω, as for a wheel, disk, or rigid wand. It is important to note that the direct proportionality of v to r is not valid for the planets, because the planets have different rotational speeds (ω).
Rotational speed can measure, for example, how fast a motor is running. Rotational speed and angular speed are sometimes used as synonyms, but typically they are measured with a different unit. Angular speed, however, tells the change in angle per time unit, which is measured in radians per second in the SI system. Since there are 2π radians per cycle, or 360 degrees per cycle, we can convert angular speed to rotational speed by:
For example, a stepper motor might turn exactly one complete revolution each second. Its angular speed is 360 degrees per second (360°/s), or 2π radians per second (2π rad/s), while the rotational speed is 60 rpm.
Rotational speed is not to be confused with tangential speed, despite some relation between the two concepts. Imagine a rotating merry-go-round. No matter how close or far you stand from the axis of rotation, your rotational speed will remain constant. However, your tangential speed does not remain constant. If you stand two meters from the axis of rotation, your tangential speed will be double the amount if you were standing only one meter from the axis of rotation.