Acceleration

Example

When a vehicle starts from a standstill (zero velocity, in an inertial frame of reference) and travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the vehicle turns, an acceleration occurs toward the new direction and changes its motion vector. The component of acceleration of the vehicle in the direction of its velocity is called a linear acceleration or tangential acceleration, its effect being the passengers on board experience as a fictitious force pushing them back into or away from their seats. When changing direction, the component of the acceleration perpendicular to the velocity is called radial or normal acceleration (or centripetal acceleration during circular motion), the reaction to which the passengers experience as a centrifugal force (another fictitious force). If the speed of the vehicle decreases, this is an acceleration in the opposite direction of the velocity vector, sometimes called deceleration or retardation, and passengers experience the reaction to deceleration as an inertial force pushing them forward. Such deceleration is often achieved by retrorocket burning in spacecraft. Both acceleration and deceleration are treated the same, as they are both changes in velocity. Each of these accelerations (tangential, radial, deceleration) is felt by passengers until their relative (differential) velocity is neutralised in reference to the acceleration due to change in speed.

Tangential and centripetal acceleration

The velocity of a particle moving on a curved path as a function of time can be written as: $${\displaystyle \mathbf{v}=v\frac{\mathbf{v}}{v}=v{\mathbf{u}}_{\mathrm{t}},}$$ with v equal to the speed of travel along the path, and $${\displaystyle {\mathbf{u}}_{\mathrm{t}}=\frac{\mathbf{v}}{v},}$$ 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 and the changing direction of u_t, 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:

$${\displaystyle \begin{array}{rl}\mathbf{a}& =\frac{d\mathbf{v}}{dt}\\ & =\frac{dv}{dt}{\mathbf{u}}_{\mathrm{t}}+v\frac{d{\mathbf{u}}_{\mathrm{t}}}{dt}\\ & =\frac{dv}{dt}{\mathbf{u}}_{\mathrm{t}}+\frac{v^2}{r}{\mathbf{u}}_{\mathrm{n}},\end{array}}$$

where u_n 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. The components $${\displaystyle {\mathbf{a}}_{\mathrm{t}}=\frac{dv}{dt}{\mathbf{u}}_{\mathrm{t}}\text{and}{\mathbf{a}}_{\mathrm{c}}=\frac{v^2}{r}{\mathbf{u}}_{\mathrm{n}}}$$ are called the tangential acceleration and the normal or radial acceleration (or centripetal acceleration in circular motion, see also circular motion and centripetal force), respectively.

Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the Frenet–Serret formulas.

Special cases

Uniform acceleration

Uniform or constant 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 ${\displaystyle {\mathbf{F}}_{\mathbf{g}}}$ acting on a body is given by: $${\displaystyle {\mathbf{F}}_{\mathbf{g}}=m\mathbf{g}.}$$

Because of the simple analytic properties of the case of constant acceleration, there are simple formulas relating the displacement, initial and time-dependent velocities, and acceleration to the time elapsed: $${\displaystyle \begin{array}{ccc}\mathbf{x}(t)& ={\mathbf{x}}_0+{\mathbf{v}}_{0}t+\frac{1}{2}\mathbf{a}{t}^2& ={\mathbf{x}}_0+\frac{1}{2}\left({\mathbf{v}}_0+\mathbf{v}(t)\right)t\\ \mathbf{v}(t)& ={\mathbf{v}}_0+\mathbf{a}t\\ v^2(t)& ={v_0}^2+2\mathbf{a}\cdot [\mathbf{x}(t)-{\mathbf{x}}_0],\end{array}}$$where

• ${\displaystyle t}$ is the elapsed time,
• ${\displaystyle {\mathbf{x}}_0}$ is the initial displacement from the origin,
• ${\displaystyle \mathbf{x}(t)}$ is the displacement from the origin at time ${\displaystyle t}$,
• ${\displaystyle {\mathbf{v}}_0}$ is the initial velocity,
• ${\displaystyle \mathbf{v}(t)}$ is the velocity at time ${\displaystyle t}$, and
• ${\displaystyle \mathbf{a}}$ is the uniform rate of acceleration.

In particular, 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, which describes, e.g., the trajectory of a projectile in vacuum near the surface of Earth.

Circular motion

In uniform circular motion, that is moving with constant speed along a circular path, a particle experiences an acceleration resulting from the change of the direction of the velocity vector, while its magnitude remains constant. The derivative of the location of a point on a curve with respect to time, i.e. its velocity, turns out to be always exactly tangential to the curve, respectively orthogonal to the radius in this point. Since in uniform motion the velocity in the tangential direction does not change, the acceleration must be in radial direction, pointing to the center of the circle. This acceleration constantly changes the direction of the velocity to be tangent in the neighbouring point, thereby rotating the velocity vector along the circle.

• For a given speed ${\displaystyle v}$, the magnitude of this geometrically caused acceleration (centripetal acceleration) is inversely proportional to the radius ${\displaystyle r}$ of the circle, and increases as the square of this speed: $${\displaystyle a_c=\frac{v^2}{r}.}$$
• For a given angular velocity ${\displaystyle \omega }$, the centripetal acceleration is directly proportional to radius ${\displaystyle r}$. This is due to the dependence of velocity ${\displaystyle v}$ on the radius ${\displaystyle r}$. $${\displaystyle v=\omega r.}$$

Expressing centripetal acceleration vector in polar components, where ${\displaystyle \mathbf{r}}$ is a vector from the centre of the circle to the particle with magnitude equal to this distance, and considering the orientation of the acceleration towards the center, yields $${\displaystyle {\mathbf{a}}_c=-\frac{v^2}{|\mathbf{r}|}\cdot \frac{\mathbf{r}}{|\mathbf{r}|}.}$$As usual in rotations, the speed ${\displaystyle v}$ of a particle may be expressed as an angular speed with respect to a point at the distance ${\displaystyle r}$ as $${\displaystyle \omega =\frac{v}{r}.}$$Thus ${\displaystyle {\mathbf{a}}_c=-{\omega }^2\mathbf{r}.}$

This acceleration and the mass of the particle determine the necessary centripetal force, directed toward the centre of the circle, as the net force acting on this particle to keep it in this uniform circular motion. The so-called 'centrifugal force', appearing to act outward on the body, is a so-called pseudo force experienced in the frame of reference of the body in circular motion, due to the body's linear momentum, a vector tangent to the circle of motion.

In a nonuniform circular motion, i.e., the speed along the curved path is changing, the acceleration has a non-zero component tangential to the curve, and is not confined to the principal normal, which directs to the center of the osculating circle, that determines the radius ${\displaystyle r}$ for the centripetal acceleration. The tangential component is given by the angular acceleration ${\displaystyle \alpha }$, i.e., the rate of change ${\displaystyle \alpha =\dot{\omega }}$ of the angular speed ${\displaystyle \omega }$ times the radius ${\displaystyle r}$. That is, $${\displaystyle a_t=r\alpha .}$$

The sign of the tangential component of the acceleration is determined by the sign of the angular acceleration (${\displaystyle \alpha }$), and the tangent is always directed at right angles to the radius vector.

Coordinate systems

In multi-dimensional Cartesian coordinate systems, acceleration is broken up into components that correspond with each dimensional axis of the coordinate system. In a two-dimensional system, where there is an x-axis and a y-axis, corresponding acceleration components are defined as $${\displaystyle \begin{array}{cc}{a}_x& =\frac{d{v}_x}{dt}=\frac{d^{2}x}{d{t}^2},\\ a_y& =\frac{d{v}_y}{dt}=\frac{d^{2}y}{d{t}^2}.\end{array}}$$ The two-dimensional acceleration vector is then defined as ${\displaystyle \mathbf{a}=⟨a_x,a_y⟩}$. The magnitude of this vector is found by the distance formula as $${\displaystyle |a|=\sqrt{a_x^2+a_y^2}.}$$ In three-dimensional systems where there is an additional z-axis, the corresponding acceleration component is defined as $${\displaystyle a_z=\frac{d{v}_z}{dt}=\frac{d^{2}z}{d{t}^2}.}$$ The three-dimensional acceleration vector is defined as ${\displaystyle \mathbf{a}=⟨a_x,a_y,a_z⟩}$ with its magnitude being determined by $${\displaystyle |a|=\sqrt{a_x^2+a_y^2+a_z^2}.}$$

Relation to relativity

Special relativity

The special theory of relativity describes the behaviour of objects travelling relative to other objects at speeds approaching that of light in 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.

General relativity

Unless the state of motion of an object is known, it is 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 equivalence principle, 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.

Conversions