Just as we talked about distance and displacement as scalar and vector quantities, we are going to introduce two new terms and relate them to concepts we know. The first is speed. Speed is defined as distance traveled per unit time and is a measure of how fast something is moving. We measure speed in any units that represent distance/time. Examples include meters/sec (m/s), miles per hour (mph), and my personal favorite, furlongs per fortnight. Note that we have not discussed a direction: we know how fast we are going but we don't know where we are going. Speed is a scalar quantity. Let's also define:
- Instantaneous Speed: Our speed at any instant. This is what the speedometer on your car measures.
- Average Speed: The total distance traveled divided by the total time. If I travel 300 kilometers in 5
hours, I may speed up, slow down, or even stop during the trip. But my average speed for the trip is 60
km/hr.
Velocity (v) is a vector quantity and is defined as the displacement per unit time. It is important to note that velocity has both a magnitude and a direction: 20 m/s down, 50 mph east, etc. We define velocity as:
where x_{f} is our final position, x_{o} is our initial position, t_{f} is our ending time, and t_{o} is our starting time.
Velocity may also be stated in terms of instantaneous velocity (v) and average velocity (v_{ave}). Instantaneous velocity is our velocity at any instant. Average velocity is the displacement per unit time, and depends solely on initial and final position. Average velocity may be found using the formula:
or by using the formula:
where v_{f} is the final velocity and v_{o} is the initial velocity. Note that this is just a straight mathematical average.
Consider two planes traveling towards each other at 60 m/s.
Both planes have the same speed: 60 m/s. But both planes have different
velocities. If I choose to call east positive and west negative, then the plane on the left has a velocity of 60m/s east or +60 m/s, while the plane on the right has a velocity of 60 m/s west, or -60
m/s. Now, recall our discussion on Reference Frames. Relative velocity is the apparent velocity of one object with respect to
another object. To the pilot in the plane on the left, in his reference frame, he is standing still but he sees a plane approaching him at -120 m/s. The pilot in the other plane sees himself at rest
and a plane approaching him at 120 m/s. Speed is constant if the
magnitude is not changing. However,
for velocity to be constant, both the magnitude and direction must be constant. A runner on a circular track may have a constant speed of 6 m/s, but his velocity is constantly changing because his
direction changes as he runs around the track.
Just as we looked at the graph of displacement vs. time, we also need to look at a graph of velocity vs. time to see what information we can glean from it. Let's start with the following displacement vs. time plot:
In our plot above, we show five intervals of a student on a journey:
- A student walks away from the origin to a displacement of 8 meters in 10 seconds.
- The student then stands still for 10 seconds.
- The student walks 4 meters towards the origin in 4 seconds.
- The student stands still for 6 seconds.
- The student walks away from the origin to a displacement of 14 meters in 5 seconds.
For each of these intervals, we can compute the slope of the line. Since slope is defined as the rise over the run, or the DY/DX, we find the slope using the formula:
But recall from above (go ahead - look up there) that DX/Dt is our definition of velocity. This brings us to our first important point: The slope of the line on a displacement vs time graph is the velocity of the object at that time. Our slopes for each of the five intervals are:
Interval Number |
Slope = velocity |
1 |
8 meters/10 seconds = .8 m/s |
2 |
0 meters/10 seconds = 0 m/s |
3 |
-4 meters/4 seconds = -1 m/s |
4 |
0 meters/6 seconds = 0 m/s |
5 |
10 meters/5 seconds = 2m/s |
So what does the graph look like?
Notice that in this case, we have both a positive and a negative y axis to show velocity away from the origin (positive) and velocity towards the origin (negative). We could also graph displacement vs. time with a similar axis of the object traveling with respect to the origin were to return back to the origin and pass the origin traveling in the opposite direction. A second important feature of a velocity vs. time graph is that the area under the curve represents the displacement for that interval, and the total area under the curve represents the total displacement. For example, in the above graph, region one is a rectangle. The area is given by height times length, or velocity times time. In this case, the area is .8 m/s times 10 seconds or 8 meters. So our total displacement in the first region is 8 meters. While this seems obvious here since we used the displacement vs. time plot to find our velocity vs. time plot, we could generate these two plots the other way around. In other words, given a plot of velocity vs. time, we could then draw our displacement vs. time plot.
Finally, let's have a quick discussion of Reaction Time. Reaction time is the amount of time that passes between noticing an event and reacting to that event. For example, if you are driving along at 20 m/s on a scenic Maine highway and you spot a moose, it takes a second or so for your brain to register the moose and your foot to hit the brakes. That time is your reaction time. Suppose it takes 1.5 seconds for this to happen. The distance you travel in that time, 20 m/s times 1.5 seconds or 30 meters is your reaction distance. Hopefully you weren't playing with your cell phone at the time because with a moose on the road, you need to stop!
Often it is helpful to see what someone else has to say on the topic. Try http://www.physicsclassroom.com/Class/1DKin/U1L1d.html for more information on speed and velocity. You should also review http://www.physicsclassroom.com/Class/1DKin/U1L3a.html for further study and problems on position vs. time graphs, and http://www.physicsclassroom.com/Class/1DKin/U1L4a.html for more information on Velocity vs. time graphs.