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## Calculus 1

### Course: Calculus 1 > Unit 4

Lesson 2: Straight-line motion- Introduction to one-dimensional motion with calculus
- Interpreting direction of motion from position-time graph
- Interpreting direction of motion from velocity-time graph
- Interpreting change in speed from velocity-time graph
- Interpret motion graphs
- Worked example: Motion problems with derivatives
- Motion problems (differential calc)
- Analyzing straight-line motion graphically
- Total distance traveled with derivatives

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# Interpreting direction of motion from position-time graph

Given the position-time graph of a linear motion, we can interpret whether the moving object moves forward or backward (or right or left, depending on how we define the motion!).

## Want to join the conversation?

- So.. Why is it, that at the peaks and valley's of a 'smooth' wave like this, is the slope zero , but at a sharp bend, the slope literally doesn't exist..? What's the difference?(4 votes)
- It simply doesn't make sense that the slope exists at a cusp. Take |x| for example; would it be -1, or 1, or 0? Why would one of these slopes take precedence over the others? Whereas with a smooth curve, the limit of the derivative at that point actually approaches a single, reasonable value.(6 votes)

- how to write equation for acceleration vs constant speed(4 votes)
- @2:35it is shown that the object is moving backward. However isn't the object at rest at that point? I am a little lost(1 vote)
- At that point we have reached our starting point again, but, as can be seen with the slope of the line at that point, we are still walking backwards. This can also be seen by the fact that 1 second after this point, we are behind the starting point.(2 votes)

- How can we say what is forward and what is backwards of the starting point? isn't that all relative?(1 vote)
- It is not exactly clear given context in the question but you may have things like vertical displacement, horizontal displacement.

I think you need just get use to the concept.(1 vote)

- what is the motion of a person at point a in the chart motion on a walk(0 votes)

## Video transcript

- [Instructor] An object
is moving along a line. The following graph gives
the object's position, relative to its starting point, over time. For each point on the graph,
is the object moving forward, backward, or neither? So pause this video and
try to figure that out. All right, so we see we
have position in meters versus time, so for example, this point right over here
tells us that after one second, we are four meters ahead
of our starting point. Or for example, this point right over here says that after four seconds,
we are almost it seems, almost four meters behind
our starting point. So let's look at each of
these points and think about whether we're moving forward,
backward, or neither. So at this point right over here, at that moment, we're
about 2 1/2 meters in front of our starting point. We're at a positive
position of 2 1/2 meters. But as time goes on,
we are moving backwards closer and closer to the starting point. So this is we are moving backward. One way to think about it, at this time we're at 2 1/2 meters. If you go forward about half a second, we are then back at our starting point. So we had to go backwards. And if we look at this
point right over here, it looks like we were going
backwards this entire time while our curve is downward sloping. But at this point right over here, when we are about it
looks like five meters behind our starting point,
we start going forward again. But right at that moment,
we are going neither forward nor backwards. It's right at that moment
where we just finished going backwards and we're
about to go forward, and one way to think about it is, what would be the slope of the
tangent line at that point, and the slope of the
tangent line at that point would be horizontal. And so this is neither. So we can use that same technique
to think about this point. The slope is positive. And we see that all right,
right at that moment, it looks like we are
at the starting point, but if you fast forward even a few, even a fraction of a
second, we are now in front of our starting point. So we are moving forward. We are moving forward right over here. And at this point, we are
at our starting point, but if you think about
what's going to happen a moment later. A moment later we're gonna
be a little bit behind our starting point, and so
here we are moving backward. And we're done.