AP Physics - Motion in one Dimension



Graphical Interpretation of 1D Motion

Position vs Time Graphs:

Let’s look at a graph of position vs time:

Displacement is plotted on the y axis and time is plotted on the x axis. The curve is a straight line.

[pic]

Therefore the slope of the displacement vs time graph is the velocity.

EXAMPLE: What is the average velocity of the object whose motion is depicted in this graph from 10 to 15 s?

[pic]

Velocity vs Time Graphs:

Since we are graphing velocity on the vertical axis, the vertical change is simply the change in velocity, or (v. We have (t for the x axis. So the slope is:

[pic]

But [pic] is the acceleration a!

Therefore the slope of the velocity vs time graph is the acceleration.

Now, look at the area between the curve (in this case, a line) and the x-axis on this graph.

In this case the area is a triangle. To calculate the area of a triangle we use the geometry formula,

[pic]

The units come out to meters.

Therefore the area of the velocity vs time graph is the displacement.

EXAMPLE: (a) During a baseball game a player runs after a fly ball. What is the player’s acceleration from a to b, b to c, and c to d? (b) what is the player’s total displacement?

Lets tackle part a first, the accelerations:

The slope of the curve represents the acceleration.

a to b: [pic]

b to c: the slope is zero, so a is zero. The player is moving at a constant speed of 4 m/s.

c to d: [pic]

The player is slowing down. He ends up moving at 2 m/s when he finally catches the ball.

(b) Now let’s figure out the displacement.

Look at that graph, it’s not a simple triangle. But, we can divide it up into simple rectangles and triangles, as shown. To calculate the total displacement, we need to find the area of each section, then add them up

After some calculations, the total displacement is the sum

of all of these displacements:

[pic]

Verify!

-----------------------

5

5

x (m)

Position Time Graph

t (s)

10

15

15

10

5

15

10

5

5

v (m/s)

t (s)

Velocity Time Graph

[pic]

5

v (m/s)

t (s)

Velocity Time Graph

d

c

15

10

5

5

b

a

x

t

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