Acceleration

In physics, acceleration is defined as the rate of change of velocity of an object with respect to time, serving as a fundamental concept in kinematics and dynamics that describes how quickly an object speeds up, slows down, or alters its direction of motion. As a vector quantity, acceleration possesses both magnitude and direction, meaning that any change in the velocity vector—whether it involves a change in speed, a change in direction, or both—constitutes an acceleration.

Definition and Mathematics

In classical mechanics, the average acceleration of an object over a period of time is defined as the change in its velocity divided by the change in time. Mathematically, this is expressed as $\mathbf{\bar{a}} = \frac{\Delta \mathbf{v}}{\Delta t}$, where $\Delta \mathbf{v}$ is the change in velocity and $\Delta t$ is the change in time.

Instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero. It is the first derivative of velocity with respect to time and the second derivative of position with respect to time, expressed in calculus as $\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2}$. While the term "deceleration" is commonly used in everyday language to describe a decrease in speed, in physics, it is simply defined as acceleration in the direction opposite to the object's current velocity.

Units of Measurement

The International System of Units (SI) defines the standard unit of acceleration as the meter per second squared (m/s²), which can also be read as "meter per second per second." This indicates the change in velocity (in meters per second) for every second that passes.

In the United States customary system, acceleration is often measured in feet per second squared (ft/s²). In the context of gravimetry and geophysics, the gal (Gal), named after Galileo Galilei, is frequently used; one gal is defined as one centimeter per second squared (1 cm/s²). Additionally, standard gravity, denoted as $g$, is a standard unit used to express accelerations in aviation, automotive engineering, and spaceflight, with $1 g$ defined as exactly 9.80665 m/s².

Components and Types of Acceleration

When an object moves along a curved path, its acceleration can be resolved into two orthogonal components: tangential acceleration and centripetal (or radial) acceleration. Tangential acceleration is parallel to the trajectory and is responsible for the change in the object's speed. Centripetal acceleration is directed toward the center of curvature of the path and is responsible for the change in the object's direction. For an object moving in a circle of radius $r$ at a constant speed $v$, the magnitude of the centripetal acceleration is given by $a_c = v^2/r$.

In rotational kinematics, angular acceleration describes the rate of change of angular velocity over time. Denoted by the Greek letter alpha ($\alpha$), it is measured in radians per second squared (rad/s²) and is related to tangential acceleration by the equation $a_t = r\alpha$, where $r$ is the radius of rotation.

Dynamics and Newton's Second Law

The relationship between acceleration, mass, and force is formalized by Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration ($\mathbf{F} = m\mathbf{a}$). This equation implies that acceleration is directly proportional to the net force applied and inversely proportional to the mass of the object. Mass, in this context, acts as a measure of an object's inertia, or its resistance to changes in its state of motion. Consequently, a larger force is required to achieve the same acceleration for a more massive object.

Acceleration in Special Relativity

In the framework of special relativity, the classical definition of acceleration requires modification when objects move at velocities approaching the speed of light ($c$). As an object's speed increases, its coordinate acceleration (the acceleration measured by an observer in a specific inertial frame) decreases for a constant applied force, ensuring that the object's velocity never exceeds the speed of light.

Relativistic mechanics also distinguishes between coordinate acceleration and proper acceleration. Proper acceleration is the physical acceleration experienced by an object itself, independent of the coordinate system, and is the quantity measured by an accelerometer attached to the object. While coordinate acceleration can be non-zero for an object in free fall under gravity when viewed from an external frame, its proper acceleration is zero, reflecting the absence of non-gravitational forces acting upon it.

Human Perception and the Equivalence Principle

Human beings do not directly perceive coordinate acceleration; rather, they perceive proper acceleration through the vestibular system in the inner ear and mechanoreceptors in the body. This is why a person in free fall, such as an astronaut in orbit, experiences weightlessness despite undergoing significant coordinate acceleration toward the Earth. The physical sensation of weight is actually the result of the normal force opposing gravity, which provides a proper acceleration.

This distinction is foundational to Albert Einstein's equivalence principle, a cornerstone of general relativity. The principle posits that the local effects of a gravitational field are indistinguishable from the effects of being in an accelerated reference frame. Thus, the acceleration due to gravity and the mechanical acceleration of a vehicle are physically equivalent in their local effects on an observer.