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## AP®︎/College Physics 1

### Course: AP®︎/College Physics 1>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 start fraction, start text, m, end text, divided by, start text, s, end text, end fraction.
Instantaneous speedSpeed at a given moment in time. Equal to the magnitude of the instantaneous velocity. Has SI units of start fraction, start text, m, end text, divided by, start text, s, end text, end fraction.
EquationSymbol breakdownMeaning in words
v, with, \bar, on top, equals, start fraction, delta, x, divided by, delta, t, end fractionv, with, \bar, on top is average velocity, delta, x is displacement, and delta, t is change in time.Average velocity is displacement divided by time interval of displacement.
v, start subscript, start text, a, v, g, end text, end subscript, equals, start fraction, d, divided by, delta, t, end fractionv, start subscript, start text, a, v, g, end text, end subscript is average speed, d is distance, and 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.
A perpendicular pair of black axes are given. The vertical axis is labeled "x (m)" and the horizontal axis is labeled "t(s)". The origin of these axes is at the lower left. A blue line extends diagonally up and to the right. On this line are two points labeled "P_1" and "P_2", with P_1 being closer to the origin. A red arrow labeled "Delta x" points upwards from P_1 . From the end of this first arrow, a second red arrow labeled "Delta t" points horizontally to point P_2. At the lower right in red is the the equation v = Delta x/Delta t.
start text, s, l, o, p, e, end text, equals, start text, v, e, l, o, c, i, t, y, end text, equals, start fraction, delta, x, divided by, delta, t, end fraction
To calculate the average velocity between two points P, start subscript, 1, end subscript and P, start subscript, 2, end subscript, we divide the change of position delta, x by the change in time delta, t.
The instantaneous velocity at point P, start subscript, 1, end subscript is equal to the slope of the position graph at point P, start subscript, 1, end subscript.

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

To find the displacement between two points P, start subscript, 1, end subscript and P, start subscript, 2, end subscript on a velocity vs. time graph, we find the area under the curve between the two points.
A perpendicular pair of black axes are given. The vertical axis is labeled "v (m/s)" and the horizontal axis is labeled "t(s)". The origin of these axes is at the lower left. A blue line extends horizontally to the right from near the top of the vertical axis. On this line are two points labeled "P_1" and "P_2", with P_1 being closer to the vertical axis. The area that is below the blue line and between the two points is shaded light pink and is labeled with the equation: Area=Delta x=v*Delta t. Directly below the point P1 is a red tick mark on the horizontal axis labeled "t1". Directly below the point P2 is a red tick mark on the horizontal axis labeled "t2". A red arrow extends from the tick mark at t1 to the tick mark at t2 and is labeled with the equation "Delta t=t2-t1".
start text, A, r, e, a, end text, equals, start text, d, i, s, p, l, a, c, e, m, e, n, t, end text, equals, v, delta, t
The change in time 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.