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### Course: High school physics (DEPRECATED)>Unit 1

Lesson 4: Velocity and speed from graphs

# Instantaneous velocity and speed from graphs review

Review the key terms and skills related to analyzing motion graphs, such as finding velocity from position vs. time graphs and displacement from velocity vs. time graphs.

## Key terms

TermMeaning
Instantaneous velocityVelocity at a given moment in time. Has SI units of $\frac{\text{m}}{\text{s}}$.
Instantaneous speedSpeed at a given moment in time. Equal to the magnitude of the instantaneous velocity. Has SI units of $\frac{\text{m}}{\text{s}}$.
EquationSymbol breakdownMeaning in words
$\overline{v}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}$$\overline{v}$ is average velocity, $\mathrm{\Delta }x$ is displacement, and $\mathrm{\Delta }t$ is change in time.Average velocity is displacement divided by time interval of displacement.
${v}_{\text{avg}}=\frac{d}{\mathrm{\Delta }t}$${v}_{\text{avg}}$ is average speed, $d$ is distance, and $\mathrm{\Delta }t$ is change in time .Average speed is distance divided by time interval for the distance traveled.

## Analyzing motion graphs

### Velocity is the slope of position vs. time graph

The equation for the slope of a position vs. time graph matches the definition of velocity exactly.
$\text{slope}=\text{velocity}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}$
To calculate the average velocity between two points ${P}_{1}$ and ${P}_{2}$, we divide the change of position $\mathrm{\Delta }x$ by the change in time $\mathrm{\Delta }t$.
The instantaneous velocity at point ${P}_{1}$ is equal to the slope of the position graph at point ${P}_{1}$.

### Displacement is the area under the curve on a velocity vs. time graph

To find the displacement between two points ${P}_{1}$ and ${P}_{2}$ on a velocity vs. time graph, we find the area under the curve between the two points.
$\text{Area}=\text{displacement}=v\mathrm{\Delta }t$
The change in time $\mathrm{\Delta }t$ will be the width of the area, and the height $v$ is on the vertical axis.

## Common mistakes and misconceptions

Some people think instantaneous velocity and speed are the same as average velocity and speed. When people use the words speed or velocity, they usually mean instantaneous velocity or instantaneous speed. Average velocity and speed account for motion occurring over a time period, and instantaneous velocity and speed describe motion at a given moment in time.

For deeper explanations of velocity and speed see the videos on instantaneous speed and velocity and position vs. time graphs.
To check your understanding and work toward mastering these concepts, check out these exercises:

## Want to join the conversation?

• can you show example and exercise how to solve instantaneous velocity or speed involving curve graph.. i very like this website to learn things that i missed in class bcs of the way of explanation process very clear, fun, and easy to understand.. i hope you can improve the 'instantaneous' topics and notify me please.. ^_^
• I have difficulty conceptualizing a displacement as an area. The math makes sense. I just can't "see" it.
• For me, I looked at it by drawing a flat line between the times on the x axis starting at the y of where the diagonal's going up📈 starts or going down📉 ends. Then if the triangle doesn't connect to the bottom you add a rectangle above or below the triangle so that it touches making sure to still keep it within the x range that the triangle above it is in. If there is a flat line, then draw a rectangle above or under it until it touches the x-axis and remains under the max time on the x-axis.
• It would have been useful if Sal had included the negative velocity aspect in his video.
• How do you you find the Instantaneous velocity speed from graphs review?
• The slope of a given point is the instantaneous velocity. For example if you had a position vs. time graph, then if you were to find the instantaneous velocity of a moment, point P, then the slope of point P is the instantaneous velocity. Was this helpful?
• Can someone tell me why the area under the slope is the displacement?
• Displacement is basically the vector form of distance, and we know distance is speed x time. So displacement is velocity (vector form of speed) x time, which is what’s plotted on the graph. The area under the graph will thus be velocity x time, giving you the displacement.
• I don't really get the concept of displacement.
What is the easiest way to figure out displacement problems?
(1 vote)
• say you walk 1 meter right, away from your house, then you walk 1 meter back to your house. you travelled 2 meters, so your distance is 2, but you ended up 0 meters from your house, so your displacement is 0.
If walk 1 meter to the left of your house, your distance is 1 meter and your displacement is either 1 or -1, depending on what you arbitrarily designate as forward (positive) or backward (negative) (the convention is to use right as forwards).
On a graph, a line going forwards is like you walking to the right, and a line going down is like going to the left. if the y is positive, you are to the right of your house, if it is negative, you are at the left of your house.
• What does it mean when it says that instantaneous speed is the magnitude of instantaneous velocity?
• It's the quantity without the sign. So if the instantaneous velocity is, for example, -2m/s, the instantaneous speed is 2m/s.
• can any one please explain to me the part,

At what time does the penguin have the same position as
t=0s?

why we need to calculate the two triangles and cancel them out then find the rest displacement ? can anyone help and thank you guys.
• The first triangle has an area of 5 from the line.
The second triangle has an area of -2.5 from the line.
That means you need to find at what point past 3 seconds will remove another 2.5.
The answer is 3.5.
That's because at 3.5 seconds a rectangle from the -2.5 triangle to the end of 3.5 is formed.
It's area is -2.5 because 5 * 0.5 = 2.5, and it's under the line so negative.