Instantaneous acceleration: Acceleration at a particular moment.

You have learned that velocity can be either average or instantaneous. Similarly, you can determine the average acceleration or the instantaneous acceleration of an object.

We use the mouse in Concept 1 on the right to show the distinction between the two. The mouse moves toward the trap and then wisely turns around to retreat in a hurry. The illustration shows the mouse as it moves toward and then hurries away from the trap. It starts from a rest position and moves to the right with increasingly positive velocity, which means it has a positive acceleration for an interval of time. Then it slows to a stop when it sees the trap, and its positive velocity decreases to zero (this is negative acceleration). It then moves back to the left with increasingly negative velocity (negative acceleration again). If you would like to see this action occur again in the Concept 1 graphic, press the refresh button in your browser.

We could calculate an average acceleration, but describing the mouse's motion with instantaneous acceleration is a more informative description of that motion. At some instants in time, it has positive acceleration and at other instants, negative acceleration. By knowing its acceleration and its velocity at an instant in time, we can determine whether it is moving toward the trap with increasingly positive velocity, slowing its rate of approach, or moving away with increasingly negative velocity.

Instantaneous acceleration is defined as the change in velocity divided by the elapsed time as the elapsed time approaches zero. This concept is stated mathematically in Equation 1 on the right.

This limit equation defines a derivative, shown as the second equation in Equation 1. Instantaneous acceleration is the derivative of velocity with respect to time. This means that if you have an expression representing the velocity of an object as a function of time, differentiating it gives you the acceleration as a function of time. Since velocity is in turn the derivative of position with respect to time, acceleration is the second derivative of position with respect to time.

Earlier, we discussed how the slope of the tangent at any point on a position-time graph equals the instantaneous velocity at that point. We can apply similar reasoning here to conclude that the instantaneous acceleration at any point on a velocity-time graph equals the slope of the tangent, as shown in Equation 2. Why? Because slope equals the rate of change, and acceleration is the rate of change of velocity.

In Example 1, we show a graph of the velocity of the mouse as it approaches the trap and then flees. You are asked to determine the sign of the instantaneous acceleration at four points; you can do so by considering the slope of the tangent to the velocity graph at each point.