Like we said before, physics is dependent on both magnitude (a number) and direction. Here, well give you definitions and examples of scalars and vectors. Its pretty common to get messed up in a mechanics problem when you get used to dealing with vector values (like displacement) and youre asked a scalar value (distance). Always be sure you understand exactly what is being asked and, as always, if youre allowed to use the equation you want to use to solve the problem. This next part is especially true in mechanics: most of the vector terms youll learn are some kind of a rate of a previous topic. Here we have velocity is a rate of change in distance, and acceleration is a rate of change in velocity. Paying attention to those relationships will better help you develop a physics mind.
Motion can be described by position, displacement, distance, velocity, speed, and acceleration.
The linear motion of a system can be described by the displacement, velocity, and acceleration of its center of mass.
Acceleration is equal to the rate of change of velocity with time, and velocity is equal to the rate of change of position with time.
The slope of the x-t graph gives you velocity.
The slope of the v-t graph gives you acceleration, and the area under the v-t graph gives you the change in displacement.
The area under the a-t graph gives you the change in velocity.
Defining & Graphing Motion
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
One of the most popular types is a position time graph. It shows an objects position as a function of time. 0861
So let us assume that we have some cute little dog that wanders away from our house at a constant 1 m/s. 0866
So the dog does that -- starts at time 0 -- wanders away from the house for a little bit, so the position is getting further and further away from this origin -- the house -- until here the dog decides it has had enough and takes a five second rest. 0872
It bops down in the grass in the backyard for 5 s. Its position does not change. 0885
As we look at the area under the graph of our velocity time graph -- if we take this area, the velocity under the graph, the area between the 0 line and where our graph is, we get that rectangle.1075
The area of that rectangle is length times width. 1089
Our length is 5 s. Our width is 1 m/s, which is 5 m. 1094
By the time we get to 5, our area is 5 m, but a look at our position time graph, right at that point, the position is 5 m. 1103
If we keep going through our graph and say "Hey over here at 8 s, what total area do we have?" 1112
All the area to the left of that 8 s is still 5 m, so over here at 8 s, our position is still 5 m. 1119
And if we keep going -- if we wanted to say what is the dog's position over here at 12 1/2 s -- well -- now we have another area to take into account. 1128
We also have this rectangle and since it is below the line -- although in math they may tell you there is not officially a negative area, there is a meaning to negative areas on the graphs in physics -- that area, which is going to be length times width -- we have 2 1/2 s width and we have -2 m/s. 1138
If we have a velocity time graph and we take the slope, we get acceleration -- or the other direction -- if we take the area under the acceleration time graph, we get the change in velocity. 1322
From the velocity time graph, if we take the area, we get the change in position. 1332
You can keep going with this pattern forward and backwards to find whatever information you need to based on these motion graphs. 1336
Let us take a look at another example with a position times, sometimes distance time graph. 1345
Which graph here best represents the motion of a block accelerating uniformly down an inclined plane? 1350
Well let us think about what is going to happen if we have some inclined plane or a ramp and we have a block that is accelerating down the ramp. 1358
Initially it is going to be at position 0 and over time it is going to get a larger and larger distance traveled. 1369
So, right away, we can eliminate number one on our choices. 1378
Now, as it goes faster and faster, it is going to cover more and more distance. 1383
We would also think of that as seeing that. . . . 1389
If we looked over here at a velocity time graph, we could make a velocity time graph and say "You know, it probably starts at some 0 velocity and goes faster and faster and faster." 1392
Well if that is the case, we also need to look at something where the area is getting progressively bigger, therefore the distance traveled must be getting progressively greater for the same time interval. 1403
Well, when the ball hits the ground, its acceleration is going to be positive for a second. 1575
It has to be in that direction to change its velocity. 1582
So we are going to have to have a spike in our acceleration time graph. 1584
For our velocity graph, it is going to start at 0 and it is going to go faster and faster and faster. 1591
Then, when it hits the ground it is going to have a spike. 1596
It is going to have a very high velocity as it sways back up, slowing down, slowing down, slowing down -- stopping. 1599
And finally, as we look at the position of the basketball -- if we take a look, we have to have some sort of path that allows the ball to do that to come back up to Bobbie's hand. 1608
So that is a pretty in-depth example and much more complicated example of how you can put position, velocity, and acceleration all together to make one complete story for what is happening to an object. 1621
Draw the velocity time graph for a ball tossed upward which returns to the point from which it was tossed. 1636
Well, I am going to start off by making my axis again. This is going to be a velocity time graph. 1643
If we toss something upwards -- like throw it up -- the moment it leaves my hand, it has its biggest velocity. Right? 1653
It's positive -- slowing down, slowing down, slowing down, slowing down -- stops for a split second, switches directions, speeds up, speeds up, speeds up, speeds up, speeds up, but in the negative direction. 1661
So if it starts off with its biggest velocity -- it could be there -- a little bit later at its highest point, for a split second it stops, then it goes faster, and faster, and faster in the opposite direction. 1672
So the velocity time graph for that situation would look something like that. 1685
All right, let us take a look at one last example. 1693
How can we get displacement from a velocity time graph? 1696
The graph below shows the velocity of an object travelling in a straight line is a function of time. 1700
Determine the magnitude of the total displacement of the object at the end of the first 6 s. 1705
So we have a velocity time graph -- we want displacement. Right away you should be thinking area. 1711
Velocity time graph wants displacement -- you need to take the area. 1717
We need the area of everything under the graph to the left of that. 1723
Again, a couple of ways you can do this -- but the easiest way I see it off the top of my head is to break this up into a triangle and a rectangle. 1729
The area under that should give us the total displacement. 1741
So we have the area of the triangle, 1/2 base times height or 1/2 times our base 2 s times our height of 10 m/s is going to be 1/2 x 2 x 10 -- 10 and seconds versus seconds in the denominator -- meters. 1745
And the area of our rectangle, length times width, or from 2 to 6 s is 4 s times its height -- 10 m/s -- seconds over seconds cancel out -- 40 m -- so the total displacement then, I just add those two up, 40 + 10 -- 50 m. 1767
Hopefully, that gets you started with some of these quantities that describe motion and motion graphs, particle diagrams, position time diagrams, velocity time diagrams, acceleration time diagrams -- gets you started, gets you going. 1791
I definitely recommend some more practice on your own. 1804
Thanks for watching Educator.com. We will be back soon. Make it a great day!1807
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