Episode 4 recaps constant velocity and uniformly accelerated motion by taking a peek at position vs time
Episode 4 recaps constant velocity and uniformly accelerated motion by taking a peek at position vs time (1:47), velocity vs time (6:48), and acceleration vs time graphs (8:30). We also look at the relevance of the slope and area of some of these graphs. How can we know an object’s displacement by looking at the velocity vs time graph? (5:00)
The Question of the Day asks: (9:30)
If you are rolling along at constant speed on your hoverboard, and you start to accelerate, which geometric shape will be made on the velocity vs time graph?
Thank you for listening to The APsolute RecAP: Physics 1 Edition!
(AP is a registered trademark of the College Board and is not affiliated with The APsolute RecAP. Copyright 2020 - The APsolute RecAP, LLC. All rights reserved.)
Website:
EMAIL:
Follow Us:
Hi and welcome to the APsolute Recap: Physics 1 Edition. Today’s episode will focus on the graphs of motion that you will create and the most important features of each of these graphs.
Lets Zoom out:
Unit 1 – Kinematics
Topic 1.2 – Representations of Motion
Big Idea - Change
If you have listened to the last episode, then you know we are talking all about 1-D motion and more specifically, the concepts of displacement, velocity, and acceleration. These are the three amigos of kinematics, OK we should probably include time, so… … the four amigos? Doesn’t have the same ring. Either way, in this episode our goal is to tie the three together.
Lets Zoom in:
It is a snowy and icy winter day, and while walking down the street you see a vehicle and driver go slowly sliding by on the level icy road. Not to worry, the driver was inside the vehicle and fortunately there were no other motorists. Wanting to practice what you have learned from The APsolute recAP Physics 1 edition, you take out your trusty stopwatch and chart the time it takes for the car to slide by each street lamp. The streetlamps are spaced uniformly 10 m apart… boy, that was convenient! According to your stopwatch, it takes the car 5 seconds to slide from the first streetlamp to the second. It takes another 5 seconds to slide to the next streetlamp, and so on. Being a good student, you decide to plot a graph of position vs. time with position in meters on the y-axis and time in seconds on the x-axis. Look at that! Wouldn’t you know it, you have a nice straight line with a positive slope.
You are curious about the slope of the line, and remember hearing a math teacher say that slope is the rise over the run. Your line seems to increase along the y-axis, or the position changes by 10 m (the rise) for every 5 seconds of time that passes along the x-axis (the run). This yields a slope of +2 once the 10/5 is reduced. However in Physics, we care strongly about the measurement units as well. And in the example we just looked at we have 10 m / 5 s so our slope is +2 m/s! Would you look at that! Sounds a lot like a velocity to me! And, you would be correct, the slope of a position vs. time graph is velocity.
This may seem like a strange thing to do, but lets graph each consecutive slope (velocity in this case) over the time. The slope of a line is constant, so it appears this graph may be easier than we thought. On the vertical axis we will graph the velocity and along the horizontal x-axis, we will graph the time. You can imagine that we now will have just a horizontal line at +2 m/s for every consecutive second. So basically, it is a line with no slope or a slope of 0. That doesn’t mean it's not an important feature to look at. In fact, we can still find the slope of the line, and its units. The rise = 0 m/s while the run = 5 s. So, the velocity of the car changed by 0 m/s / 5 s and would you look at that! The slope of the velocity vs time graph is 0 meter per second… per second, or m/s2. And, just like that, we can see that the sliding car maintained its velocity, never speeding up or slowing down, so the acceleration of the sliding car is 0 m/s2 for every second that passed.
But wait… there’s more! You notice that you stopped recording data after 25 seconds of time, and when you plotted your +2 m/s velocities in the second graph, you created a familiar geometric shape. You have a rectangle. “That’s pretty cool!” you say to yourself. “I wonder what the area of that rectangle is?” So, you grab your trusty pencil, and then realize it has a broken point, throw it in the trash, and reach for a more reliable mechanical pencil and begin shading in the rectangle. You of course know that the area of a rectangle is length times width and you come up with 2 x 25 so an area of 50. 50 what!?!?!? Well, we had one side of the rectangle which was the velocity so it was +2 m/s and then we had the length of the rectangle along the x-axis which was 25 seconds, and we multiplied m/s by s so we got… … +50 m? Wait, a displacement? YOU GOT IT! The area bound by the velocity vs time graph is in fact the displacement. Whoa! That seems useful!
Bored of watching cars slide along the icy road by yourself, you decide to go skiing with your friend. Still amazed at your discoveries earlier with the sliding car, you decide to measure your friend’s position over time as you both slide down the treacherous precipice known as the bunny hill. This time, things turn out quite differently. Luckily, the owners of the ski resort have installed fencing that has posts every 1m. Man, this is getting scary! Your friend began from rest when she set off down the hill. At 1 second your friend is by the 1st marker (1m). After 2 seconds, she is cruising past the 4th marker (4 m), at 3 seconds she is at the 9th marker, and by 4 seconds she has arrived at the 16th marker. You plot her position vs. time and realize that it looks like a quadratic graph, the right side of a parabola opening up. Well shucks! That seems a lot more complicated.
You do however notice that the slope began pretty flat, slightly sloping in a positive direction, and as you got to the 4 second portion, the graph was much steeper and still positively sloped. You think to yourself, “Maybe if I graph velocity, I can get something useful.” For reasons a bit more complex than we want to get into here, instead of grabbing a straight edge to find consecutive slopes, you look at your cell phone’s GPS data for velocity and luckily it tracked the last few seconds as you went down the hill with your friend. It recorded 0 m/s, 2 m/s, 4 m/s, 6 m/s, and 8 m/s for each of the 4 seconds you traveled down the hill.
You decide to graph these velocities vs. time, and notice that the slope is not zero like with the sliding car, but rise (+2 m/s) over run (1 s) gives you a slope of +2 m/s every 1 s. That is the acceleration! +2 m/s2. You also notice that when you draw an invisible line straight up from 4 seconds you have created a geometric shape, but this time you don’t see a rectangle, you see a triangle. Well… that’s different! “I still know how to find the area of that shape, it is just ½ base times height.” you say to your friend who by now looks a lot less interested, and just wants to go to the lodge to get hot chocolate. The base seems to be the 4 seconds, and the height seems to be the +8 m/s final velocity we reached. So, ½*(4)(8) = 16… oh right, 4 s and +8 m/s, so +16 m. Wait… a… tick! That’s how far we went!
Finally, you decide to graph your +2 m2 acceleration over the 4 second time period and you create an a-t graph (acceleration vs time). The graph has a horizontal line at +2 m/s/s for the entire 4 seconds, and… … NO WAY! It makes another rectangle! This time the rectangle has sides +2 m/s/s * 4 seconds or an area of 8 m/s, and that’s how much our velocity changed! It turns out the area of an acceleration vs. time graph is equivalent to the change in velocity. Ok, now I need some hot chocolate too!
To recap……
The slope of a position vs. time graph is the velocity, the slope of a velocity vs. time graph is acceleration, the area bound by the velocity vs. time function graph and the x-axis is the displacement, and the area bound by the acceleration vs. time function and the x-axis is the change in velocity.
Coming up next on the APsolute RecAP Physics 1 Edition: 1-D kinematic equations and some problem solving.
Today’s Question of the day is about graphing kinematic equations.
Question: If you are rolling along at constant speed on your hoverboard, and you start to accelerate, which geometric shape will be made on the velocity vs time graph?