Section 1.6 and 1.7 in Matter and Interactions (4th edition)
Predicting or explaining motion often requires you to use some sort of representation (or visual aid). A common (and incredibly useful) one is the graph. In these notes, you read about graphs of motion and how to translate between different graphs.
You want to track the motion (as a function of time) of a car that is moving at constant velocity. For this situation, the object's motion can be predicted exactly using the position update formula:
→rf=→ri+→vavgΔt
While the motion of the car, in principle, can occur 3 dimensions, it's not possible to represent all three dimensions and the time variable on a single 2-D graph. So, we have to select a component of the car's position (or velocity) to plot. In this case, let's assume the car moves to the right (i.e., in the +x direction). Perhaps, the plot of the car's position vs time looks like this:
Here, you can see that the position of the car changes linearly with time, as we would predict for a car moving at constant velocity. From this graph, you can also determine the car's initial position (12 m), final position (132 m), and average velocity (12 m/s).
For this motion, the average velocity is the same as the instantaneous velocity. Recall, the definition of the average (→vavg) and instantaneous velocity (→v) are:
→vavg=Δ→rΔt
Both definitions are connected to the idea of slope. It might be easier to see this from the average velocity in one-dimension where the rise is the change in position and the run is the change in time:
vavg,x=slopeofpositionvstimegraph=riserun=ΔxΔt
For any graph, this will give the average slope between any two points. For the case above, it gives the same number for the slope between any two points and at any point. The slope at a given point is given by evaluating the derivative of the function at that point. You can think of this as taking smaller and smaller “runs”; that is, by letting Δt go to zero. In that case, you have determined the instantaneous velocity:
vx=instantaneousslope=dxdt