3.6:
Instantaneous Acceleration
The acceleration of an object at any given instant is called instantaneous acceleration. It is the limit of the average acceleration as the time interval approaches zero. It is the first derivative of velocity with respect to time.
Consider the example of a woman walking on a road from point P1 to P2. At point P1, she has a velocity of v1x at time t1. And after some time at t2, her velocity is v2x at point P2. The change in her velocity is given by Δv in a time interval of Δt.
As we consider smaller intervals of time – when the point P2 approaches P1, in the limit when Δt tends to zero, instantaneous acceleration at point P1 is given by the slope of the tangent to the curve at point P1.
Lastly, the instantaneous acceleration of a body can also be calculated as the second derivative of the position with respect to time using the position versus time graph.
3.6:
Instantaneous Acceleration
Acceleration is in the direction of the change in velocity, but it is not always in the direction of motion. When an object slows down, its acceleration is opposite to the direction of its motion. Although commonly referred to as deceleration, this causes confusion in our analysis as deceleration is not a vector, and does not point to a specific direction with respect to a coordinate system. Therefore, the term deceleration is not used. For example, when a subway train slows down, it accelerates in a direction opposite to its direction of motion. In other words, acceleration is in the negative direction of the chosen coordinate system, so it is said that the train is undergoing negative acceleration. If an object in motion has a velocity in the positive direction with respect to a chosen origin and it acquires a constant negative acceleration, the object eventually comes to a rest and reverses direction.
Instantaneous acceleration, or the acceleration at a specific instant in time, is obtained by the same process as instantaneous velocity—that is, by considering an infinitesimal interval of time. For example, to find instantaneous acceleration using only algebra, we must choose an average acceleration that is representative of the motion.
This text is adapted from Openstax, University Physics Volume 1, Section 3.3: Average and Instantaneous Acceleration.