Name:
Date:
Motion Graphing– Lab 3
Aim: What information can we gather from graphing and analyzing a function?
Background
What is the independent and dependent variable and why are they called that?
Hypothesis
Materials
Procedures
1. Read the background information.
2.
3.
Be sure to graph one linear and one curved graphs.
Section:
Data and
Analysis
My Station is number _____________
My Data
Direction
Time
Notes
The Class Data
Time at Run #
stations ( )
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Run #
( )
Run #
( )
Run #
( )
I will graph Run _______ and Run ________ because
Slope calculations:
Run #
( )
Run #
( )
Slope calculations:
Conclusion
What information can we gather by taking the slope of the displacement time graph? What
about the slope of the velocity time graph?
Motion Graphing
We'll use the letter m for slope.
Here's the official formula:
So, in our last example...
But, notice something cool...
The order of the points doesn't matter! Let's switch them and see what we get:
Recall the area or a rectangle is L x W (length x width) and the area of a
triangle is ½ bh ½ (base x height)
Equations can easily contain the information equivalent of several sentences. Galileo's description of an object
moving with constant speed (perhaps the first application of mathematics to motion) required one definition,
four axioms, and six theorems. All of these relationships can now be written in a single equation.
or
Equations are great for describing idealized situations, but they don't always cut it. Sometimes you need a
picture to show what's going on — a mathematical picture called a graph. Graphs are often the best way to
convey descriptions of real world events in a compact form. Graphs of motion come in several types depending
on which of the kinematic quantities (time, displacement, velocity, acceleration) are assigned to which axis.
Displacement-time
Let's begin by graphing some examples of motion at a constant velocity.
Three different curves are included on the graph to the right, each with an
initial displacement of zero. Note first that the graphs are all straight. (Any
kind of line drawn on a graph is called a curve. Even a straight line is called a
curve in mathematics.) This is to be expected given the linear nature of the
appropriate equation. (The independent variable of a linear function is raised
no higher than the first power.)
On a displacement-time graph…

slope equals velocity.

the "y" intercept equals the initial displacement.

when two curves coincide, the two objects have the same displacement at that time.

straight lines imply constant velocity.

curved lines imply acceleration.

an object undergoing constant acceleration traces a portion of a parabola.

average velocity is the slope of the straight line connecting the endpoints of a curve.

instantaneous velocity is the slope of the line tangent to a curve at any point.

positive slope implies motion in the positive direction.

negative slope implies motion in the negative direction.

zero slope implies a state of rest.
Velocity-time
The most important thing to remember about velocity-time graphs
is that they are velocity-time graphs, not displacement-time graphs.
There is something about a line graph that makes people think they're
looking at the path of an object. A common beginner's mistake is to
look at the graph to the right and think that the the v = 9.0 m/s line
corresponds to an object that is "higher" than the other objects. Don't
think like this. It's wrong.
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. In
these graphs, higher means faster not farther. The v = 9.0 m/s line is
higher because that object is moving faster than the others.
These particular graphs are all horizontal. The initial velocity of
each object is the same as the final velocity is the same as every
velocity in between. The velocity of each of these objects is constant
during this ten second interval.
On a velocity-time graph…

slope equals acceleration.

the "y" intercept equals the initial velocity.

when two curves coincide, the two objects have
the same velocity at that time.

straight lines imply uniform acceleration.

curved lines imply non-uniform acceleration.

an object undergoing constant acceleration traces a straight line.

average acceleration is the slope of the straight line connecting the endpoints of a curve.

instantaneous acceleration is the slope of the line tangent to a curve at any point.

positive slope implies an increase in velocity in the positive direction.

negative slope implies an increase in velocity in the negative direction.

zero slope implies motion with constant velocity.

the area under the curve equals the change in displacement.
acceleration-time
The acceleration-time graph of any object traveling with a constant velocity is
the same. This is true regardless of the velocity of the object. An airplane
flying at a constant 600 mph (270 m/s), a sloth walking with a constant speed
1 mph (0.4 m/s), and a couch potato lying motionless in front of the TV for
hours will all have the same acceleration-time graphs — a horizontal line
collinear with the horizontal axis. That's because the velocity of each of these
objects is constant. They're not accelerating. Their accelerations are zero. As
with velocity-time graphs, the important thing to remember is that the height
above the horizontal axis doesn't correspond to position or velocity, it
corresponds to acceleration.
If you trip and fall on your way to school, your acceleration towards the
ground is greater than you'd experience in all but a few high performance cars
with the "pedal to the metal". Acceleration and velocity are different quantities.
Going fast does not imply accelerating quickly. The two quantities are
independent of one another. A large acceleration corresponds to a
rapid change in velocity, but it tells you nothing about the values of the
velocity itself.
On an acceleration-time graph…

slope is meaningless.

the "y" intercept equals the initial acceleration.

when two curves coincide, the two objects have the same acceleration at that time.

an object undergoing constant acceleration traces a horizontal line.

zero slope implies motion with constant acceleration.

the area under the curve equals the change in velocity.