*Class content I* > *The Main Question: Motion > Kinematics > Kinematic Variables *> *Acceleration*

3.2.2.2

**Prerequisites**

Average acceleration tells us the change in velocity over some time interval. If we want to know the acceleration at a particular time, we have to make that time interval small. We still need two times in order to see a change in velocity, but if the time change is small enough, we can identify the acceleration as belonging to the time in the middle of the very small time interval.

When our Δ*t* is small enough, we identify the acceleration at that (central) time as the *instantaneous acceleration* and as the derivative of the velocity:

Notice that since v is the derivative of the position, the instantaneous acceleration is the second derivative of the position.

When we "open up" the vectors, we see that this vector equation stands for two statements about the acceleration: the x-acceleration is the derivative of the x-velocity, and the y-acceleration is the derivative of the y-velocity.

Joe Redish 9/10/11

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