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A runner is at the position $x=0 \mathrm{m}$ when time $t=0$ s. One hundred meters away is the finish line. Every ten seconds, this runner runs half the remaining distance to the finish line. During each ten-second segment, the runner has a constant velocity. For the first forty seconds of the motion, construct (a) the position-time graph and $(b)$ the velocity-time graph.
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Physics 101 Mechanics
Chapter 2
Kinematics in One Dimension
Motion Along a Straight Line
Cornell University
University of Michigan - Ann Arbor
Hope College
McMaster University
Lectures
04:34
In physics, kinematics is the description of the motion of objects and systems in the frame of reference defined by the observer. An observer has to be specified, otherwise the term is meaningless.
07:57
In mathematics, a position is a point in space. The concept is abstracted from physical space, in which a position is a location given by the coordinates of a point. In physics, the term is used to describe a family of quantities which describe the configuration of a physical system in a given state. The term is also used to describe the set of possible configurations of a system.
07:36
A runner is at the positio…
06:48
A slowing race Starting at…
04:09
Two runners start one hund…
04:03
05:37
0:00
04:10
A running model A model fo…
02:39
Races The velocity functio…
08:06
okay for this problem. We're drawing a position versus time graph of a runner who's running for 40 seconds to try and complete 100 meter race. And every 10 seconds The amount of the race like finish is half of what's remaining. So we're gonna draw kind of a sketch of this position versus time graph. So on my x axis, I have the time. And on the Y axis, I have my position. Now, since this race is going to, ah 100 meters, gonna mark the top of my position graph as 100 meters. Um, and all the way at the beginning. Down here, we have zero meters, so I'm gonna put some time intervals. We have 10 meters or 10 seconds in 20 seconds in 30 seconds in and 40 seconds in at the end of the race and gonna kind of draw these up here to keep us focused on what we're looking at. We need to know the 10 the 20 the 30 and the 40. Now what the problem tells us is that every 10 seconds, the runner completes half of what's remaining in the race. So initially, the runner starts at zero and there is Ah, 100 meters left in the race. So the runner is going to go half of the way there and travels actually the full 50 meters Now, in the next 10 seconds, the runner slows down a little bit and only covers half of what's remaining. So half of 50 gives us 25 meaning this runner is only gonna make it to the 75 meter mark over the next 10 seconds. So losing some steam. Then there's 25 meters left. So the runner travels only half of that as well. And we had 87.5 right here and then with the 87.5 mark left, there's only 12.5 meters left. So they're under finishes half of that and finishes at 93.75 for that final point here. And that's 93 0.75 meters. So the position versus time graph because this is all constant velocity. We don't know anything about any accelerations other than they abruptly change. For each of these 10 seconds, we're going to connect my points now. Most likely, this is going to be a constant acceleration curve where they're slowing down. Is there running? But we don't know enough to do that. So we're connecting it with the straight lines here. Um, now, looking at this graph, we can see that the runner is slowing down over time because the slope of this graph is getting less and less and the slope of a position versus time graph is equal to its velocity. Um, so I'm going to draw our velocity versus time graph now, um, and kind of opposite how you would think about it as our concepts get more complicated, like velocity is a more complicated concept than that. Displacement on the grafts get simpler. So since the slope of that position versus time graph is equal to our velocity, um, the first section where we were going 10 seconds 2030 40 that first section where we were just going 10 seconds, um, to find the velocity, we would do the slope of that line. So the rise of 50 divided by the time of 10 gives us a velocity of five meters per second. Then the second chunk, they only go from 50 to 75. So it's going half of that distance in the same 12th span, so the velocity from 10 to 20 drops to 2.5 meters per second. The next 10 seconds sees that cut in half again to 1.25 meters per second and the last piece gets cut in half, one more time to 0.6 to 5 meters per second. Now this graph is not very realistic because we have kind of these gaps where the velocity abruptly changes and that's probably not what's actually happening. Um, as you move forward with this concept, you'll get into what acceleration grafts looked like and that would add us another layer to look at. But until then, this is a good approximation of what the position versus time graph looks like. And with the velocity versus
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