# Acceleration

by Ron Kurtus (updated 9 January 2023)

The * acceleration* of an object is its change in velocity over an increment of time. Since velocity is a vector (indicating magnitude and direction), acceleration is also a vector.

If the magnitude and/or direction of the velocity changes, so too does that of the acceleration. If the velocity is slowing down, the object is said to decelerate or having negative acceleration.

Acceleration units can be meters per second squared (m/s

^{2}) or feet per second squared (ft/s^{2}).

Considerations about acceleration include calculating average acceleration and instantaneous acceleration, as well as looking at uniform acceleration in a straight line and circular motion.

Some questions you may have include:

- What is average acceleration?
- What is instantaneous acceleration?
- What are examples of constant acceleration?

This lesson will answer those questions. Useful tool: Units Conversion

## Average acceleration

Average acceleration is the change of velocity over a period of time.

When a moving automobile accelerates in a straight line, it goes from one velocity to another. Its average acceleration is the change in velocity divided by the time taken to make that change:

a_{av}= Δv/Δt

where

**a**is the average acceleration (m/s_{av}^{2}or ft/s^{2})**Δv = v**is the change in velocity (_{2}- v_{1}**Δ**is the Greek letter capital delta)**Δt = t**is the time interval for_{2}- t_{1}**Δv**

Deceleration is when a moving object slows down. It is also called negative acceleration.

### Straight line acceleration

If an object is moving in a straight line, its acceleration is the difference of its velocity or speed along that line divided by the time increment.

### Direction change acceleration

When a moving object changes direction, it is accelerating.

If **v _{1}** is the initial velocity in the

**x**-direction, and

**v**is the velocity after changing directions, the acceleration is found by breaking

_{2}**v**into its

_{2}**x**and

**y**components and dividing by the time increment.

Let **v _{x}** be the component of

**v**along the same axis as

_{2}**v**, and let

_{1}**v**be the component in the perpendicular axis. Then the resulting accelerations in the two directions are:

_{y}

aand_{x}= (v_{x}− v_{1})/Δta_{y}= v_{y}/Δt

Square the components, add them together, and take the square root. The resulting acceleration due to a change in direction is:

a = √(a_{x}^{2}+ a_{y}^{2})

## Instantaneous acceleration

Instantaneous acceleration is the change of velocity over an instance of time.

Instantaneous acceleration is defined as the limit of the averageÂ accelerationÂ when the interval of time considered approaches 0. It is also defined in a similar manner as the derivative of velocity with respect to time. It is the rate at which velocity changes at a specific instant in time.

The equation for instantaneous acceleration is:

a_{in}= lim_{(Δt=0)}Δv/Δt

where **lim(Δt=0)**** Δv/Δt** is the limit of **Δv/Δt** as **Δt** approaches 0

It is usually written as:

a = dv/dt

where

**dv**is the derivative of**v****dt**is the derivative of**t**

The derivative of **v** is the limit as **Δv** approaches **0**. Likewise for **dt **is the limit as **Δt** approaches **0**. Derivatives are commonly used in Calculus.

## Constant acceleration

Constant or uniform acceleration is when the velocity changes the same amount in every equal time period.

One example of uniform acceleration concerns the acceleration due to gravity for falling objects. In such a case, the magnitude of the velocity increases at a constant rate, but its direction is constant.

Another example of constant acceleration is when the magnitude of the velocity is constant, but the direction changes at a constant rate. This can be seen when swinging a weight attached to a string in a uniform circular motion.

## Summary

The acceleration of an object is its change in velocity over an increment of time. This can mean a change in the object's speed or direction

Average acceleration is the change of velocity over a period of time. Instantaneous acceleration is the change of velocity over an instance of time.

The acceleration due to gravity and uniform circular motion are examples of constant or uniform acceleration.

Strive to do your best

## Resources and references

### Websites

**Acceleration** - Physics Hypertextbook

**Acceleration** - The Physics Classroom

**Acceleration** - Wikipedia

### Books

(Notice: The *School for Champions* may earn commissions from book purchases)

**Top-rated books on the Physics of Motion**

## Students and researchers

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**www.school-for-champions.com/science/
motion_acceleration.htm**

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## Where are you now?

## Acceleration