The velocity is a vector quantity and is the speed with a direction. Acceleration exists when there is a change in speed–either by speeding up or slowing down–or change in direction or both changing simultaneously.

The mathematical way of expressing acceleration is the change in velocity over time.

Assume an object is moving at a constant speed. While making a turn, it slows down. Its instantaneous velocity changes both in magnitude and direction. Let the instantaneous velocity vectors located at P₁ and P₂ be vectors v₁ and v₂.

By subtracting the vectors, the change in velocity, delta vector v, is obtained.

Now, the direction of the average acceleration vector will be in the direction of the change in velocity. The average acceleration vector is given by the ratio between the Δ vector v and Δt.

When we consider smaller intervals of time, the position P₂ approaches P₁. At this instant, as Δt approaches zero, the average acceleration approaches instantaneous acceleration.

In everyday conversation, accelerating means speeding up. Acceleration is a vector in the same direction as the change in velocity, Δv, therefore the greater the acceleration, the greater the change in velocity over a given time. Since velocity is a vector, it can change in magnitude, direction, or both. Thus acceleration is a change in speed or direction, or both. For example, if a runner traveling at 10 km/h due east slows to a stop, reverses direction, and continues their run at 10 km/h due west, their velocity has changed as a result of the change in direction, even though the magnitude of the velocity is the same in both directions. Thus, acceleration occurs when velocity changes in magnitude (an increase or decrease in speed), direction, or both.

The acceleration vector is the instantaneous acceleration, and it can be obtained from the derivative of the velocity function with respect to time. Acceleration varies greatly with different objects and has nothing to do with the size of an object or its mass. Acceleration can also vary significantly with time during the motion of an object. For example, a drag racer has a large acceleration just after its start, but then it tapers off as the vehicle reaches a constant velocity. Its average acceleration can be quite different from its instantaneous acceleration at a particular time during its motion.

This text is adapted from Openstax, University Physics Volume 1, Section 4.2: Acceleration Vector.

4.3 : Acceleration Vectors

Motion in Two or Three Dimensions

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