 # Graphical Analysis of Motion.

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Graphical Analysis of Motion

Which car/s is/are undergoing an acceleration?

Which car experiences the greatest acceleration?

Match a Graph Consider the position-time graphs below. Each one of the 3 lines on the position-time graph corresponds to the motion of one of the 3 cars. Match the appropriate line to the particular color of car.

Section 1: Position vs Time Graphs

Position vs Time Graphs
position (x) vs time (t) graphs plot position as a function of time the x-t graph of a constantly moving object is linear

If an object is staying in the same position for a time interval, then it’s…
at rest. stopped. not moving. still. Represented by a horizontal line on a position vs time graph

velocity- displacement traveled in a certain amount of time; a vector quantity
can be constant or average constant velocity- velocity that does not change

Constant Velocity

Constant Velocity Questions: What is slope?
What is the formula for slope? Use the slope formula to determine the slope of the line. What do you notice?

Constant Velocity Conclusions:
the slope of a position vs time graph is the velocity m = rise = ∆y = displacement = v run ∆x time

b c a d e

Questions: Where does the graph have positive slope? What does this positive slope mean? Where does the graph have negative slope? What does this negative slope mean? Is there any place(s) on the graph where the object is not moving? How do you know? What is the slope of the graph at this section?

Average Speed vs Average Velocity
Velocity is not the same as speed. average velocity- the change in position divided by the time it took for that change to occur; a ratio of an object’s change in position (displacement) to the time it takes for that change to occur vavg = distance traveled time of travel vavg = ∆x ∆t

Speed or Velocity? speed & velocity are different quantities
speed is the rate distance covered over time speed is a scalar quantity velocity is the rate displacement traveled over time velocity is a vector quantity speed & velocity equal magnitudes when an object does not change direction

Average Velocity Consider a car moving with a constant velocity of +10 m/s for 5 seconds. The diagram below depicts such a motion.

Average Velocity The position-time graph would look like the graph below:

How would this graph look?
Average Velocity Now consider a car moving at a constant velocity of +5 m/s for 5 seconds, abruptly stopping, & then remaining at rest (v = 0 m/s) for 5 seconds. How would this graph look?

Average Velocity

Instantaneous Speed speed at any given point in time

Can I use one point to find velocity?
Important Note: DO NOT use one point to calculate velocity on a position-time graph!!! b c a d e

Example 2—Drawing Graphs from Graphs
Use the position vs time below to draw a velocity vs time graph.

Solution Velocity vs Time 6 5 4 3 2 1 -12 -8 -4 12 8 Velocity (m/s)
12 8 Velocity (m/s) Time (s)

Position vs Time Graphs
the x-t graph of a constantly moving object is linear

Position vs Time Graphs
the slope of an x-t graph is rise (position) over run (time) for an x-t graph, m = ∆x ∆t

Position vs Time Graphs
the change in position of an object is displacement, ∆x the rate change in position of an object over an interval of time (the time it takes for the change to occur) is called average velocity, vavg vavg = ∆x ∆t

Position vs Time Graphs

Position vs Time Graph

Position vs Time Graphs
the slope of an x-t graph describes the velocity of an object in motion Fast, Rightward(+) Constant Velocity Slow, Rightward(+) Constant Velocity

Position vs Time Graphs
the slope of an x-t graph describes the velocity of an object in motion Slow, Leftward(-) Constant Velocity Fast, Leftward(-) Constant Velocity

Position vs Time Graphs

Position vs Time Graphs

Position vs Time Graphs

Section 2: Velocity vs Time Graphs

Velocity vs Time Graphs
a velocity (v) vs time (t) graph plots velocity as a function of time the v-t graph of a constantly moving object is a horizontal line remember, horizontal lines have zero slope

Velocity vs Time Graphs
the v-t graph of an object moving at a changing rate is a diagonal line the velocity is changing by a constant rate

Constant, Uniform Acceleration
Bessie took a trip on her bicycle. The data of Bessie’s bike ride below, along with its accompanying graph, is shown. Use the graph to answer the questions that follow:

Constant, Uniform Acceleration

Constant, Uniform Acceleration
Now, here is the data, along with the resulting velocity vs time graph, of the same bike ride.

Constant, Uniform Acceleration
Questions: What can you say about the slope of this graph? What does that mean?

Constant, Uniform Acceleration
Conclusions: If an object has constant velocity, then that object has zero acceleration. Constant velocity  Δv equals zero. The graph of zero acceleration is a horizontal line. All horizontal lines have zero slope because Δy equals zero.

Constant, Uniform Acceleration

Velocity vs Time Graphs
the slope of a v-t graph describes changing velocity over time m = ∆v ∆t

Velocity vs Time Graphs
the rate at which velocity changes over an interval time is known as acceleration, aavg mathematically, aavg = ∆v ∆t

Velocity vs Time Graphs
the slope of a v-t graph describes the acceleration of the object in motion Positive Velocity Positive Acceleration

Velocity vs Time Graphs
units of acceleration: m/s/s, m/s2 acceleration is a vector quantity acceleration describes a change in velocity in the positive direction acceleration also describes a change in direction deceleration (negative acceleration) describes a change in velocity in the opposite direction

Velocity vs Time Graphs

Velocity vs Time Graphs

Velocity vs Time Graphs

Velocity vs Time Graphs

Section 2a: Constant Acceleration

Example 2—Drawing Graphs from Graphs
Use the position vs time below to draw a velocity vs time graph.

Solution Velocity vs Time 6 5 4 3 2 1 -12 -8 -4 12 8 Velocity (m/s)
12 8 Velocity (m/s) Time (s)

Solution Acceleration vs Time 6 5 4 3 2 1 -12 -8 -4 12 8
12 8 Example 3—Drawing Graphs from Graphs Acceleration (m/s2) Time (s)

Constant, Uniform Acceleration
velocity changes by the same rate ex: a ball rolling down a ramp

Constant, Uniform Acceleration

Constant, Uniform Acceleration
Questions: What is slope? What is the formula for slope? Use the slope formula to determine the slope of the line. What do you notice?

Constant, Uniform Acceleration
Conclusions: the slope of a velocity vs time graph is acceleration m = rise = ∆y = change in velocity = a run ∆x time

Constant, Uniform Acceleration

Section 3: Using Tangent Lines

Draw a straight line that is tangent to the curve.
Using Tangent Lines What does the following graph show? Draw a straight line that is tangent to the curve. What does tangent mean?

Position vs Time Position (m) Time (s)

Using Tangent Lines What does this graph show?

Constant Acceleration vs Constant Velocity

positive slope  positive acceleration
position vs time graph with increasing speed & its resulting velocity vs time graph: positive slope  positive acceleration

negative slope  negative acceleration
position vs time graph with decreasing speed & its resulting velocity vs time graph: negative slope  negative acceleration

Animations

Animations

Animations

Animations

Animations

Animations

Example: Determine what is happening at each section
constant velocity increasing acceleration deceleration acceleration

Section 4: Lines That Intersect

Lines That Intersect Sometimes, a problem will ask you to compare/contrast the motion of 2 objects. To do this, use the information provided in the problem to construct a position vs time graph. Then, use the graph to compare/contrast the 2 objects’ motions.

Lines That Intersect

What do the 2 cars have in common?

Do they ever share the same velocity?

Example 2—100 meter dash Harry gives Sam a 30. m head start in the 100. m dash. Harry can run at 10.0 m/s, while Sam only runs at 6.0 m/s.

Example Does Harry win?

Solution Draw a position vs time graph for a 100. m dash, plotting both runners. Position vs Time Position (m) Time (s) 100 50 5 10

Example If Harry wins, at what time & place does he catch up to Sam?

Solution y = mx + b Now, you can determine the equation of each line.
Remember, the equation of a line is Since we are using position, time, & velocity, the equation becomes y = mx + b

Solution x = vt + b

Solution xH = vHtH + bH For Harry… Where, vH = Harry’s velocity
bH = where Harry begins xH = vHtH + bH

Solution So… xH = 10t

Solution xS = vStS + bS For Sam… Where, vS = Sam’s velocity
bS = where Sam begins xS = vStS + bS

Solution So… xS = 6t + 30

Solution Since you are looking for the point where Harry & Sam pass each other, that means the coordinates (tH, xH) & (tS, xS) are equal. So, set the 2 equations for each line equal to each other & solve for t.

Solution 10t = 6t + 30 4t = 30 t = 7.5 sec

Solution xH = 10t = (10.0 m/s) (7.5 s) xH = 75 m
Now, that you know at what time Harry catches up to Sam, you can calculate at what place this happens. Plug the value for time (7.5 sec) into the equation & solve for x. xH = 10t = (10.0 m/s) (7.5 s) xH = 75 m

Section 5: Displacement from V-T Graphs

Displacement from a Velocity vs Time Graph
in the given v-t graph, the area under the graph is that for a rectangle area = l* w = v (t2 – t1) = ∆x

Displacement from a Velocity vs Time Graph
in the given v-t graph, the area under the graph is that for a rectangle ∆x