ǻv = change in instantaneous velocity
ǻt = elapsed time
What is the mouse’s average
acceleration?
2.10 - Instantaneous acceleration
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.
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.
Instantaneous acceleration
Acceleration at a particular moment