Speed and Velocity
Speed and velocity are explored more fully in another article. The points needed to understand acceleration are summarized.
In physics speed and velocity have distinct meanings. Velocity includes direction. Speed does not. A quantity that includes direction is a vector, and one that does not is a scalar. Velocity is a vector; speed is a scalar. Understand this distinction.
The average velocity is the change in position divided by the change in time. The instantaneous velocity is the limit as the time interval approaches zero of the change in position divided by the change in time. Using calculus, it is the derivative of the position with respect to time.
Average Acceleration
Acceleration is our way of measuring the rate of change of an object's velocity. Its units are a distance unit divided by a time unit squared, such as meters per second squared.
The average acceleration is defined as the change in velocity divided by the change in time. The change in velocity can be an increase in speed, a decrease in speed, or a change in direction. In equation form it is:
(average acceleration) = (change in velocity)/(change in time)
Instantaneous Acceleration
Just as for velocity, physicists also define the instantaneous acceleration.
The average acceleration over a small time interval will be closer to the true instantaneous acceleration than the average over a larger time interval. In general the smaller the time interval used for finding the average acceleration, the closer the average acceleration is to the true instantaneous acceleration. Ideally the instantaneous acceleration is the average acceleration over a time interval of 0 seconds. However that would require dividing by zero in the equation above.
Physicists therefore find the instantaneous acceleration by looking at the average acceleration over successively smaller time intervals. As the time interval approaches zero, the average acceleration approaches the instantaneous acceleration. This process is the mathematical process of taking the limit.
The instantaneous acceleration is therefore the limit as the change in time approaches 0 of the change in velocity divided by the change in time. For those familiar with calculus, this limit is the definition of the derivative. The instantaneous acceleration is therefore the derivative of the velocity with respect to time, or a=dv/dt.
Direction Change
Acceleration is defined in terms of velocity rather than speed. Therefore acceleration includes changing the direction of motion as well as changing the speed. Usually, when the acceleration is in the same direction as the velocity, it is positive and the speed increases. A negative acceleration is usually opposes the velocity, and the speed decreases. An acceleration perpendicular to the velocity changes the direction. Beginning physics students often have difficulty grasping this fact. To convince yourself it is really true, consider the following thought experiment.
When a car is stopped and the driver slams on the accelerator, the people in the car feel pushed back into their seats. Similarly when the driver slams the brakes, the occupants not wearing seatbelts fly forward. No force is pushing these occupants. The car is accelerating, but the people's inertia tends to keep them moving at the same velocity. Hence, they feel apparent, called inertial, forces because they are in the accelerating car. Now imagine the car careening around a curve at a high constant speed. The people in the car will slide across their seats. Why? The car is accelerating. Changing the car's direction is an acceleration just as much as changing its speed.
Any change in velocity is an acceleration.
Further Reading
Knight, R.D., Physics for Scientists and Engineers, Pearson, 2004.
Paul A. Heckert - I have a Ph.D. in astrophysics, over 30 years experience teaching physics and astronomy, and over 60 published research articles.
