# Relationship between slope and speed

### Graphs of Motion – The Physics Hypertextbook

The average slope between two points in time will give you the average velocity between those two points in time. The instantaneous velocity does not have to. Slope as Velocity. In this particular It is the same velocity if the slope of the graph line is the same. (, ) .. Relationship Between Displacement and Time. Thus velocity corresponds to slope and initial displacement to the intercept on the vertical The relation between displacement and time is quadratic when the.

So let me draw a slightly different one where the velocity is changing. So let me draw a situation where you have a constant acceleration. The acceleration over here is going to be one meter per second, per second. So one meter per second, squared. And let me draw the same type of graph, although this is going to look a little different now.

So this is my velocity axis. I'll give myself a little bit more space. I'm just going to draw the magnitude of the velocity, and this right over here is my time axis. So this is time. And let me mark some stuff off here. So one, two, three, four, five, six, seven, eight, nine, ten.

**Position Time Graph to Acceleration and Velocity Time Graphs - Physics & Calculus**

And one, two, three, four, five, six, seven, eight, nine, ten. And the magnitude of velocity is going to be measured in meters per second. And the time is going to be measured in seconds. So my initial velocity, or I could say the magnitude of my initial velocity-- so just my initial speed, you could say, this is just a fancy way of saying my initial speed is zero. So my initial speed is zero.

So after one second what's going to happen? After one second I'm going one meter per second faster.

### Meaning of Slope for a v-t Graph

So now I'm going one meter per second. After two seconds, whats happened? Well now I'm going another meter per second faster than that. After another second-- if I go forward in time, if change in time is one second, then I'm going a second faster than that. And if you remember the idea of the slope from your algebra one class, that's exactly what the acceleration is in this diagram right over here.

The acceleration, we know that acceleration is equal to change in velocity over change in time. Over here change in time is along the x-axis. So this right over here is a change in time. And this right over here is a change in velocity.

When we plot velocity or the magnitude of velocity relative to time, the slope of that line is the acceleration. And since we're assuming the acceleration is constant, we have a constant slope. So we have just a line here. We don't have a curve. Now what I want to do is think about a situation.

Let's say that we accelerate it one meter per second squared. And we do it for-- so the change in time is going to be five seconds. And my question to you is how far have we traveled? Which is a slightly more interesting question than what we've been asking so far. So we start off with an initial velocity of zero.

And then for five seconds we accelerate it one meter per second squared. So one, two, three, four, five. So this is where we go. This is where we are. So after five seconds, we know our velocity. Our velocity is now five meters per second.

## What are position vs. time graphs?

But how far have we traveled? So we could think about it a little bit visually. We could say, look, we could try to draw rectangles over here. Maybe right over here, we have the velocity of one meter per second.

So if I say one meter per second times the second, that'll give me a little bit of distance. And then the next one I have a little bit more of distance, calculated the same way. I could keep drawing these rectangles here, but then you're like, wait, those rectangles are missing, because I wasn't for the whole second, I wasn't only going one meter per second.

So I actually, I should maybe split up the rectangles. I could split up the rectangles even more. So maybe I go every half second.

- What are velocity vs. time graphs?
- Graphs of Motion
- How is the slope of a line related to speed?

So on this half-second I was going at this velocity. And I go that velocity for a half-second. Velocity times the time would give me the displacement.

And I do it for the next half second. Same exact idea here. Gives me the displacement. It is possible to be accelerating and yet not be moving but only for an instant, of course. Some interpret this as motion in reverse, but is this generally the case? Well, this is an abstract example.

## Why distance is area under velocity-time line

It's not accompanied by any text. Graphs contain a lot of information, but without a title or other form of description they have no meaning. What does this graph represent? A mote of dust? About all we can say is that this object was moving at first, slowed to a stop, reversed direction, stopped again, and then resumed moving in the direction it started with whatever direction that was.

Negative slope does not automatically mean driving backward, or walking left, or falling down.

The choice of signs is always arbitrary. About all we can say in general, is that when the slope is negative, the object is traveling in the negative direction. On a displacement-time graph… positive slope implies motion in the positive direction. There is something about a line graph that makes people think they're looking at the path of an object.

Don't think like this. Don't look at these graphs and think of them as a picture of a moving object. Instead, think of them as the record of an object's velocity.