Class content > The Main Question: Motion > Kinematics
Prerequisites
Just like velocity is the rate of change of position, acceleration is the rate of change of velocity. Since velocity requires a comparison of two positions -- which means paying attention to the position at two different times, acceleration really requires paying attention to the velocity at two different times -- which means considering the position at three different times.
To see why we need three positions to construct an acceleration, consider a video of a ball being thrown upward. Let's try to figure out the acceleration at frame 37 of the video. To do that, we need the velocity a little before and a little after. By comparing the positions at frames 36 and 37 we can infer a velocity at frame 36.5 (halfway between the two frames). By comparing the positions at frames 37 and 38 we can infer a velocity at frame 37.5. By comparing the velocities at frames 36.5 and 37.5 we can infer an acceleration halfway between them -- at frame 37. This is simpler in equations. Instead of considering position, velocity, and acceleration as functions of time, let's use frame number to specify the time. So y(37) means the y-coordinate of the ball in frame 37; v(36.5) means the velocity of the ball half-way between frames 36 and 37. Using v= Δy/Δt and a = Δv/Δt, and taking the time between one frame and the next to be Δt, we have
v(36.5) = [x(37) - x(36)]/Δt
v(37.5) = [x(38) - x(37)]/Δt
a(37) = [v(37.5) - v(36.5)]/Δt = = [x(38) - 2x(37) + x(36)]/Δt
It's interesting to think about why it comes out like that. If the velocity were constant (zero acceleration), then how would the position change between frames 36-37 compare with the change from 37-38? What would that mean for the frame differences for a?
Just as with velocity we will introduce two different kinds of acceleration -- the average acceleration and the instantaneous acceleration. The average acceleration is what we use when we are explicitly paying attention to the time interval; the instantaneous acceleration is what we use when the time interval we use to calculate the acceleration from the position (or the velocity) is very small compared to any times we want to pay attention to.
In either case, the acceleration answers the questions: how fast are you changing your velocity?
Read the two follow-ons for the details of average and instantaneous acceleration.
Follow-ons
Joe Redish 9/6/11
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