Motion

A Descriptive Model for Particles

AP Physics 2023-24 with Mr. Porter

Essential Questions

Does your object change its velocity?

If it does change velocity, does the velocity change constantly?

• What evidence do you need to collect to answer these questions?

Some New Tools

Video Upload

1. Record video with you phone
2. Sign into Pivot and Upload Video
3. Scale to something in video to known size
4. Use Built in tools

Tips:

• Take video perpendicular to motion
• Keep video device as still has possible (i.e. put on something, don't hold in hands)

Spark Timer

1. Tape spark tape to object
2. Pick time setting
3. Turn on timer & object and send it

Does your object change its velocity?

If it does change velocity, does the velocity change constantly?

• What evidence do you need to collect to answer these questions?

Finding Velocity from Position time Graphs

Reminder: the slope at a specific time on a position vs. time graph is the instantaneous velocity of an object

How do we find the slope of a curve?

Finding Velocity from Position time Graphs

Pivot & Rate of Change

Pivot will calculate these slopes for you with its rate of change function. We can accomplish it with a calculated column

NOTE: This only works when time intervals are the same (i.e. time between samples is 0.1 s)

How does the Rate of Change function work?

Calculates slope over three data points and assigns that value to the middle time.

Create a Velocity vs. Time Graph from your Data

From your Velocity vs. Time Graph...

Does your object change its velocity?

If it does change velocity, does the velocity change constantly?

• What does the slope represent?
• What does the intercept represent?

What about Position and Time?

What is the relationship betwen position and time?

How do we know what curve to pick??

Linearization

Finding the linear approximation of a function

Why?

1. Verify a certain relationship (i.e. determine if your selected curve fit is the relationship)
2. You expect your data has a certain relationship and want to determine a parameter.
3. You are trying to find an unknown relationship between two variables.

Linearizing allows us to interpret

curves as lines

How do we change a curve into a line?

By making a test plot

1. Make a new data column
2. Calculate the expected relationship in that column (i.e. if the graph looks quadratic calculate )
3. Make a new plot (i.e. graph vs. )
4. Analyze new linear model

Cheat Sheet

Lab Summary

• Slope of the Velocity vs. Time graph is acceleration
• Acceleration is measured in m/s/s or m/s
• Slope of the Position vs. Time is of the acceleration

General Mathematical Models:

• For this lab (with no initial position or initial velocity):

• For an object with an initial velocity and initial starting point:

Defintions & Notes

• Displacement indicates how far an object ends up from its initial position, regardless of its total distance traveled.
• Average velocity is displacement divided by the time interval over which that displacement occurred. Slope of position time graph over a time duration.
• Instantaneous velocity is how fast an object is moving at a specific moment in time. Slope of position time graph at a specific time.
• Acceleration tells how much an object’s speed changes in one second.
  • When an object speeds up, its acceleration is in the direction of motion.
  • When an object slows down, its acceleration is opposite the direction of motion.

Thinking about graphical relationships

Position-time graphs

• To determine how far from the detector an object is located, look at the vertical axis of the position-time graph.
• To determine how fast an object is moving, look at the steepness (i.e. the slope) of the position-time graph.
• To determine which way the object is moving, look at which way the position-time graph is sloped.
• A position-time slope like a front slash / means the object is moving away from the detector.
• A position-time slope like a back slash \ means the object is moving toward the detector
• Instantaneous velocity is found by taking the slope of the tangent line to a position-time graph

Velocity-time graphs

• To determine how fast an object is moving, look at the vertical axis of the velocity-time graph.
• To determine which way the object is moving, look at whether the velocity-time graph is above or below the horizontal axis.
• An object is moving away from the detector if the velocity-time graph is above the horizontal axis.
• An object is moving toward the detector if the velocity-time graph is below the horizontal axis.
• To determine how far an object travels, determine the area between the velocity-time graph and the horizontal axis.
• On a velocity-time graph it is not possible to determine how far from the detector the object is located.
• Most everyday motion can be represented with straight segments on a velocity-time graph.

Acceleration

• Negative Acceleration does not always mean slowing down
• Negative Acceleration means the velocity is changing in the negative direction

Acceleration: Increasing vs. Decreasing Speed

• Speeding Up: Acceleration and velocity have the same sign
• Slowing Down: Acceleration and Velocity have opposite signs

Shapes of Graphs - Summary

Physics is more than plugging numbers into equations!

Solve

A poorly tuned car accelerates from rest to a speed of 28 m/s in 20 seconds. What is the acceleration of the car? How far does the car travel in this time?

Remember to:

1. Sketch and Translate
2. Simplify and Diagram
3. Represent Mathematically
4. Solve + Evaluate

Hint:

Use a graph or graphs to solve the problem.

Multiple Representations:

• Verbal: Describes the motion of the car in a narrative manner.
• Pictorial: A picture shows the initial and final motion of the car.
• Diagrammatic: The picture is simplified and shows the motion of the car over multiple clock readings.
• Graphical: Three representations that show how the position, velocity, and acceleration of the car vary over time.
• Mathematical: Describes the velocity of the car as a function of time and the position of the car as a function of time.

Problem Solving Process:

1. Sketch and Translate
2. Simplify and Diagram
3. Represent Mathematically
4. Solve + Evaluate

1. Sketch

• A sketch of the object or objects at the beginning of the motion, at the end of the motion, and any important points in between where the motion changes (i.e. if the acceleration of the object changes).
• A coordinate system Label the axes and the origin (position 0).
• Symbols that represent the time, position, and velocity of each object at each moment you have draw above. Use subscripts to depict different times and different objects (i.e. or or ). This step defines the symbols that you will use later in calculations.
• Draw arrows to show the acceleration and velocity of the object between each position (or write or )

1. Translate

Now that you have a sketch of the situation you should organize everything you know and want to find. Therefore you should:

• Make a table listing all of the symbol values that are known for the problem or that can be determine from simple geometry. Check that the signs are consistent with your sketch and its coordinate system. Also list known relationships, such as , even if you don't know the values yet. At this point do any unit conversions that are necessary.
• Make a list of the unknown quantity or quantities you'll need to answer the question.

2. Simplify and Diagram (Physics Diagrams)

• From our sketch we will no create physics diagrams...for motion this includes:
  • position vs. time graph
  • velocity vs. time graph (most useful)
  • acceleration vs. time graph

3. Represent Mathematically

• mathematical description of the process
• turn physics diagrams into equations:
  • use area of v-t to calc displacement
  • use slopes to calculate velos, or accelerations
  • use position functions to find final positions

4. Solve + Evaluate

• substitute in given quantities and solve!
• Evaluate: does this number make sense? Is it a reasonable solution?

Example:

A poorly tuned car accelerates from rest to a speed of 28 m/s in 20 seconds. What is the acceleration of the car? How far does the car travel in this time?

Sketch and Translate

Simplify and Diagram

Represent Solve:

Kinematic Equations

...from AP Reference Table

• the subscript indicates the plane of motion, these equations also work for and planes of motion
• we will often remove that subscript when writing equations involving problems with only one plane of motion: i.e.

Kinematic Equations

Reminder:

so...

• becomes
• becomes

Kinematic Cross Diagram for Problem Solving

Kinematic Cross Diagram

Identifying Equations

Non-reference Table Equations

Using the Cross Diagram

1. Setup
   1. Write down the cross
   2. Write down the known variables and underline those variables in the cross
   3. Star (or circle) the variable that is unknown and is being solved for
2. Planning
   1. Circle the variables that are involved in this problem
   2. Use the equation that involves those four variables
3. Execution
   1. Solve the problem
   2. You can circle twice: solve for first unknown to use in second equation

Example

A poorly tuned car accelerates from rest to a speed of 28 m/s in 20 seconds. What is the acceleration of the car? How far does the car travel in this time?

More Complicated Example

Gideon is South's best speed skater. He accelerates from 0.00 m/s to 7 m/s in 5 s and then continues at this constant speed for another 8 s. What is the total distance Gideon skates?

Practice

A speedboat increases its speed from 14.3 m/s to 31.1 m/s in a distance of 274 m. Determine the time over which this acceleration occurs.