Understanding of Velocity and Acceleration

Velocity

The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction of motion (e.g. 60 km/h to the north). Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.

Velocity is a physical vector quantity; both magnitude and direction are needed to define it. The scalar absolute value (magnitude) of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI (metric system) as metres per second (m/s or $ms^{-1}$). For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector. If there is a change in speed, direction or both, then the object has a changing velocity and is said to be undergoing an acceleration.

Acceleration

In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the orientation of the net force acting on that object. The magnitude of an object's acceleration, as described by Newton's Second Law, is the combined effect of two causes:

the net balance of all external forces acting onto that object — magnitude is directly proportional to this net resulting force; that object's mass, depending on the materials out of which it is made — magnitude is inversely proportional to the object's mass. The SI unit for acceleration is metre per second squared $ms^{-2}$, ${\tfrac {\operatorname {m} }{\operatorname {s} ^{2}}}$).

Average acceleration

Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time t is found in the limit as time interval Δt → 0 of Δv/Δt

An object's average acceleration over a period of time is its change in velocity $(\Delta \mathbf {v} )$ divided by the duration of the period $(\Delta t)$. Mathematically,

${\bar {\mathbf {a} }}={\frac {\Delta \mathbf {v} }{\Delta t}}.$

Instantaneous acceleration

Instantaneous acceleration, meanwhile, is the limit of the average acceleration over an infinitesimal interval of time. In the terms of calculus, instantaneous acceleration is the derivative of the velocity vector with respect to time:

$\mathbf {a} =\lim _{{\Delta t}\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {d\mathbf {v} }{dt}}$

As acceleration is defined as the derivative of velocity, v, with respect to time t and velocity is defined as the derivative of position, x, with respect to time, acceleration can be thought of as the second derivative of x with respect to t:

$\mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {x} }{dt^{2}}}$

(Here and elsewhere, if motion is in a straight line, vector quantities can be substituted by scalars in the equations.)

By the fundamental theorem of calculus, it can be seen that the integral of the acceleration function a(t) is the velocity function v(t); that is, the area under the curve of an acceleration vs. time (a vs. t) graph corresponds to velocity.

$\mathbf {v} =\int \mathbf {a} \ dt$

Likewise, the integral of the jerk function j(t), the derivative of the acceleration function, can be used to find acceleration at a certain time:

$\mathbf {a} =\int \mathbf {j} \ dt$

Resources

• What is acceleration?, Khan Academy

Licensing

Content obtained and/or adapted from:

• Velocity, Wikipedia under a CC BY-SA license
• Acceleration, Wikipedia under a CC BY-SA license