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Welcome to

Trina Merrick

MCHS

*Slides/material thanks to

Dr. Peggy Bertrand of Oak Ridge High School, Oak Ridge,TN

Kinematics

- Kinematics is the branch of mechanics that describes the motion of objects without necessarily discussing what causes the motion.
- We will learn to describe motion in two ways.
- Using graphs
- Using equations

Particle

- A particle is an object that has mass but no volume and occupies a position described by one point in space.
- Physicists love to turn all objects into particles, because it makes the math a lot easier.

Position

- How do we represent a point in space?
- a) One dimension
- b) Two dimensions
- c) Three dimensions

(x)

(x,y)

(x,y,z)

Distance (d)

- The total length of the path traveled by a particle.
- “How far have you walked?” is a typical distance question.
- SI unit:
meter (m)

Displacement (Dx)

- The change in position of a particle.
- “How far are you from home?” is a typical displacement question.
- Calculated by…
x = xfinal – xinitial

- SI unit:
meter (m)

Delta ( )

- is a Greek letter used to represent the words “change in”. x therefore means “change in x”. It is always calculated by final value minus initial value.

Practice Problem

Question: If x is the displacement of a particle, and d is the distance the particle traveled during that displacement, which of the following is always a true statement?

- d = |Dx|
- d < |Dx|
- d > |Dx|
- d > |Dx|
- d < |Dx|

Practice Problem

A particle moves from x = 1.0 meter to x = -1.0 meter.

- What is the distance d traveled by the particle?
- What is the displacement of the particle?

2.0 m

-2.0 m

100 m

displacement

50 m

distance

A

Distance vs Displacement- A picture can help you distinquish between distance and displacement.

Practice Problem

You get on a ferris wheel of radius 20 meters at the bottom. When you reach the top on the first rotation

- what distance have you traveled?
- what is your displacement from the bottom?
- When you are on your way back down, does the distance increase, decrease, or stay the same? What about the displacement?
- What is the distance traveled after you have completed the full ride of 10 rotations? What about the displacement?

Practice Problem answers

You get on a ferris wheel of radius 20 meters at the bottom. When you reach the top on the first rotation

- d = ½ (2 r) = r = 20 m
- x = 20 + 20 = 40 m
- distance increases, displacement decreases
- d = 10 (2 r) = 400 m

Average Speed

- How fast a particle is moving.
- save = d
t

where:

save = rate (speed)

d = distance

t = elapsed time

- SI unit:
m/s

Average speed is always a positive number.

Average Velocity

- How fast the displacement of a particle is changing.
- vave = ∆x
∆t

where:

vave = average velocity

∆x = displacement

∆t = change in time

- SI unit:
m/s

Average velocity is + or – depending on direction.

Demonstration

- You are a particle located at the origin.
- Demonstrate how you can move from x = 0 to x = 10.0 with an average speed of 0.5 m/s. You may not leave the x-axis!
- What was your average velocity in this case?

Demonstration

- You are a particle located at the point x = 10.0 m.
- Demonstrate how you can move from x = 10.0 to x = 0 with an average speed of 0.5 m/s. You may not leave the x-axis!
- What is your average velocity in this case?

Demonstration

- You are a particle located at the origin.
- Demonstrate how you can move from x = 0 to x = 10.0 and back with an average speed of 0.5 m/s. You may not leave the x-axis!
- What was your average velocity in this case?

3.33 m/s

Practice ProblemA car makes a trip of 1½ laps around a circular track of diameter 100 meters in ½ minute. For this trip

a) what is the average speed of the car?

b) what is its average velocity?

Practice Problem

How long will it take the sound of the starting gun to reach the ears of the sprinters if the starter is stationed at the finish line for a 100 m race? Assume that sound has a speed of about 340 m/s.

Answer: 0.29 s

t

Practice ProblemDescribe the motion of this particle.

It is moving at constant velocity in the + x direction.

x

A

Dx

Dt

t

Practice ProblemWhat physical feature of the graph gives the constant velocity?

The slope, because Dx/Dt is rise over run!

vave = Dx/Dt

Force Concept Inventory

No scratch paper or calculator is necessary.

Use pencil on BLUE side of scantron sheet.

Name: Write your NAME followed by your TEST NUMBER.

Subject:FCI

Date:8/17/05

Period:???

When you are done, bring your scantron sheet to the front of the room and quietly begin working on tonight’s homework.

Practice

Q 7: Is it possible for a car to circle a race track with constant velocity? Can it do so with constant speed?

Q 8: Friends tell you that on a recent trip their average velocity was +20 m/s. Is it possible that their instantaneous velocity was negative at any time during the trip?

P 13: The human nervous system can propagate nerve impulses at about 102 m/s. Estimate the time it takes for a nerve impulse generated when your finger touches a hot object to travel the length of your arm. (HINT: How long is your arm, approximately?)

Average Velocity Lab

- Purpose: Figure out a way to make your cart move with an average velocity of as close to 0.200 m/s as possible. Use only the equipment provided. Photogate must be in PULSE mode.
- Tonight:Type your BRIEF and PARTIAL lab report. The sections I want you to do are:
- Procedure
- Data (include a tableof data for 5 trials, a sample calculations, and a diagram of your setup). Clearly indicate what you predicted your average velocity to be, and what it actually was during the demo.
- Analysis (where did your errors come from?)

t

Practice ProblemDoes this graph represent motion at constant velocity?

No, since there is not one constant slope for this graph.

x

B

Dx

Dt

t

Practice ProblemCan you determine average velocity from the time at point A to the time at point B from this graph?

Yes. Draw a line connecting A and B and determine the slope of this line.

vave = Dx/Dt

Practice Problem

You drive in a straight line at 10 m/s for 1.0 hour, and then you drive in a straight line at 20 m/s for 1.0 hour. What is your average velocity?

Answer: 15 m/s (this is probably what you expected!)

Practice Problem

You drive in a straight line at 10 m/s for 1.0 km, and then you drive in a straight line at 20 m/s for another 1.0 km. What is your average velocity?

Answer: 13.3 m/s (this is probably NOT what you expected!)

Always use the formula for average velocity; don’t just take an “average” of the velocities!

Instantaneous Velocity

- The velocity at a single instant in time.
- Determined by the slope of a tangent line to the curve at a single point on a position-time graph.

Dx

Dt

t

Instantaneous Velocityvins = Dx/Dt

Draw a tangent line to the curve at B. The slope of this line gives the instantaneous velocity at that specific time.

B

Practice Problem

The position of a particle as a function of time is given by the equation

x = (2.0 m/s) t + (-3.0 m/s2)t2.

- Plot the x vs t graph for t = 0 until t = 1.0 s.
- Find the average velocity of the particle from t = 0 until t = 0.50 s.
- Find the instantaneous velocity of the particle at t = 0.50 s.

Practice

Q 10: If the position of an object is zero, does its speed need to be zero?

Q 11: For what kind of motion are the instantaneous and average velocities equal?

P 27: The position of a particle as a function of time is given by x = (-2.0 m/s) t + (3.0 m/s2) t2.

a) Plot x-vs-t for time from t = 0 to t = 1.0 s.

b) Find the average velocity of the particle form t = 0.15 s to t = 0.25 s.

c) Find the average velocity from t = 0.19 s to t = 0.21 s.

Acceleration (a)

- Any change in velocity is called acceleration.
- The sign (+ or -) of acceleration indicates its direction.
- Acceleration can be…
- speeding up
- slowing down
- turning

Uniform (Constant) Acceleration

- In Physics B, we will generally assume that acceleration is constant.
- With this assumption we are free to use this equation:
a = ∆v

∆t

- SI Unit:
m/s2

Acceleration has a sign!

- If the sign of the velocity and the sign of the acceleration is the same, the object speeds up.
- If the sign of the velocity and the sign of the acceleration are different, the object slows down.

Practice Problem

A 747 airliner reaches its takeoff speed of 180 mph in 30 seconds. What is its average acceleration?

Practice Problem

A horse is running with an initial velocity of 11 m/s, and begins to accelerate at –1.81 m/s2. How long does it take the horse to stop?

t

Practice ProblemDescribe the motion of this particle.

It is moving in the +x direction at constant velocity. It is not accelerating.

t

Practice ProblemDescribe the motion of this particle.

It starts from rest and accelerates in the +x direction. The acceleration is constant.

v

A

Dv

Dt

t

Practice ProblemWhat physical feature of the graph gives the acceleration?

The slope, because Dv/Dt is rise over run!

a = Dv/Dt

Practice Problem

Determine the displacement of the object from 0 to 4 seconds.

Ans: 0

Describe the motion.

The object is initially moving in the negative direction at –20 m/s, slows gradually and momentarily is stopped at 2.0 seconds, and then accelerates in the + direction. At 4.0 seconds, it is back at the origin, and continues to accelerate in the + direction.

Position vs Time Graphs

- Particles moving with no acceleration (constant velocity) have graphs of position vs time with one slope. The velocity is not changing since the slope is constant.
- Position vs time graphs for particles moving with constant acceleration look parabolic. The instantaneous slope is changing. In this graph it is increasing, and the particle is speeding up.

Uniformly Accelerating Objects

- You see the car move faster and faster. This is a form of acceleration.
- The position vs time graph for the accelerating car reflects the bigger and bigger Dx values.
- The velocity vs time graph reflects the increasing velocity.

Position vs Time Graphs

- This object is moving in the positive direction and accelerating in the positive direction (speeding up).
- This object is moving in the negative direction and accelerating in the negative direction (speeding up).
- This object is moving in the negative direction and accelerating in the positive direction (slowing down).

v

a

t

t

t

Acceleration

vs

time

Position

vs

time

Velocity

vs

time

Draw Graphs forStationary Particlesv

a

t

t

t

Acceleration

vs

time

Position

vs

time

Velocity

vs

time

Draw Graphs forConstant Non-zero Velocityv

a

t

t

t

Acceleration

vs

time

Position

vs

time

Velocity

vs

time

Draw Graphs for ConstantNon-zero AccelerationPractice Problem

What must a particular Olympic sprinter’s acceleration be if he is able to attain his maximum speed in ½ of a second?

In some problems, estimation is an important part of the problem!

Practice Problem

A plane is flying in a northwest direction when it lands, touching the end of the runway with a speed of 130 m/s. If the runway is 1.0 km long, what must the acceleration of the plane be if it is to stop while leaving ¼ of the runway remaining as a safety margin?

Kinematic Equations

- v = vo + at
- Use this one when you aren’t worried about x.

- x = xo + vot + ½ at2
- Use this one when you aren’t worried about v.

- v2 = vo2 + 2a(∆x)
- Use this one when you aren’t worried about t.

Practice Problem

On a ride called the Detonator at Worlds of Fun in Kansas City, passengers accelerate straight downward from 0 to 20 m/s in 1.0 second.

- What is the average acceleration of the passengers on this ride?
- How fast would they be going if they accelerated for an additional second at this rate?
- Sketch approximate x-vs-t, v-vs-t and a-vs-t graphs for this ride.

Practice Problem

Air bags are designed to deploy in 10 ms. Estimate the acceleration of the front surface of the bag as it expands. Express your answer in terms of the acceleration of gravity g.

Practice Problem

You are driving through town at 12.0 m/s when suddenly a ball rolls out in front of you. You apply the brakes and decelerate at 3.5 m/s2.

- How far do you travel before stopping?
- When you have traveled only half the stopping distance, what is your speed?
- How long does it take you to stop?
- Sketch approximate x-vs-t, v-vs-t, a-vs-t graphs for this situation.

Practice problems

- 39. Landing with a speed of 115 m/s and traveling due south, a jet comes to rest in 7.00 x 102 m. Assuming the jet slows with constant acceleration, find the magnitude and direction of its acceleration.

Practice problems

- 40. When you see a traffic light turn red you apply the brakes until you come to a stop. If your initial speed was 12 m/s, and you were headed due west, what was your average acceleration during braking?
- 41. Suppose the car in the previous problem comes to rest in 35 m. How much time does this take?

Practice problems

- 42. Starting from rest, a boat increases its speed to 4.30 m/s with constant acceleration
- (a) What was the boat’s average speed?
- (b) If it takes the boat 5.00 s to reach this speed, how far has it traveled?

Practice problems

- 43. A cheetah accelerates from rest to 25 m/s in 6.2 s. Assuming constant acceleration,
- (a) how far has the cheetah run in this time?
- (b) How far has the cheetah run in 3.1 s?

Lab Report Analysis

- The GOODprocedure
- The BADprocedure
- The UGLYprocedure
- The POETIC procedure
- LAB REPORT FORMAT

Lab

- The Physics 500
- PartI: Determine your speed (use multiple trials, etc.) while doing two activities using a meterstick and stopwatch.
PartII: After choosing which of your “activities” is more reliable, surrender your meterstick and use you and your activity speed to determine the unknown distance marked off by your teacher.

Labs

Tumble Buggy Lab

- TumbleBuggy Lab AP Physics
- Constant Velocity
- Devise a method to determine the speed of your TumbleBuggy.
- Determine the Speed of the TumbleBuggy.
- Devise a method to determine the length of the hall USING THAT INFORMATION! The TumbleBuggy cannot, however enter the hallway. You also MAY NOT measure the hallway with a meterstick!!
- In the open area (hallway) devise a method to construct a table of data of position versus time for your tumblebuggy. You are restricted to usingt he materials on the materials table. All of them may not be needed. (Tape, paper, post-it notes, paper clips)
- Construct a Position vs. Time graph for the tumblebuggy.
- Using information from that graph, determine the speed of the tumblebuggy. Compare to the speed to the data from Part II.
- Use that information to graph velocity versus time for the tumblebuggy. Determine the acceleration from the graph.
- Plot acceleration versus time for the tumblebuggy.

Lab

- Picket Fence Lab
- Purpose: To determine an approximate value for the gravitational acceleration constant.
- Theory: The gravitational acceleration constant, g, is approximately 9.8 m/s2 near the surface of the earth. Objects in free-fall therefore accelerate toward the earth at a rate of 9.8 meters per second per second. The downward instantaneous velocity of a freely falling object follows the following equation: v = v0 - gt. If the instantaneous velocity at various points during free-fall can be determined, the gravitational acceleration constant should be able to be estimated.
- Equipment:
- Photogate Stopwatch Meterstick “Picket Fence” CBL
- Discussion: How did the acceleration you observe compare to the actual acceleration due to gravity? What assumptions did we make that could account for the differences? What are some possible sources of human and equipment error?

Lab Free Fall Times

- Purpose: To investigate the relationship between the distance an object falls from rest with the time it takes to travel that distance.
- Theory: In the case of falling from rest, the second kinematic equation
- x = xo + vot + ½ a t2
- can be use to derive the free fall equation
- y = -1/2 g t2
- where g is the acceleration due to gravity (9.8 m/s2), t is the time, and y is the distance fallen.
- Equipment: Photogate timers, droppable objects, meter sticks.
- Procedure: Come up with a method to test the validity of the free fall equation using the equipment given. Must include the dropping of more than one type of object from several different heights.
- Data and calculated results: Must appear in a clear and neat table, and must include a comparison of calculated and measured results. You might find the following equation handy:
- % difference = measured result – theoretical result
- theoretical result
- Conclusion/Discussion: Should include problems encountered in devising the procedure, a comparison of the free fall characteristics of different objects (the same or different) and a comparison of the calculated and measured results. Can you think of any errors that you might have encountered and explain how these errors might have affected your results?

Model problem (HW 44)

- A kid slides down a hill on a toboggan (a HAT?) with an acceleration of 3.0 m/s2. If he starts from rest, how far has he traveled in
- (a) 1.0 s?
- (b) 2.0 s?
- (c) 3.0 s?

Model Problem (47)

- Two car drive on a straight highway. At time t = 0, car A passes mile marker 0 traveling due north with a speed of 28.0 m/s. At the same time, car B is 2.0 km south of mile marker 0 traveling at 30.0 m/s due south. Car A is speeding up with an acceleration of magnitude 1.5 m/s2, and car B is slowing down with an acceleration of magnitude 2.0 m/s2. Write x-vs-t equation of motion for both cars.

Model Problem (48)

- A 1-ton baby elephant jumps onto the roof of a Volkswagon. Upon impact, the elephant’s speed is 5.0 m/s. The elephant makes a dent in the roof of the Voltswagon that is 50 cm deep. What is the magnitude of the elephants deceleration, assuming it is constant.

Model Problem (49)

- Superman leaps into the air and moves straight upward with constant acceleration. After 5 seconds, Superman has reached a height of 2,000 m.
- A) What is Superman’s acceleration?
- B) What is his speed at this time?

Model Problem (55)

- A yacht cruising at 2.0 m/s is shifted into neutral. After coasting 8.0 m, the engine is engaged again and the yacht resumes cruising at a reduced speed of 1.5 m/s. How long did it take the yacht to coast the 8.0 m?

Free Fall

- Occurs when an object falls unimpeded.
- Gravity accelerates the object toward the earth the entire time it rises, and the entire time it falls.
- a = -g = -9.8 m/s2
- Acceleration is always constant and toward the center of the earth!!!

Symmetry in Free Fall

- When something is thrown upward and returns to the thrower, this is very symmetric.
- The object spends half its time traveling up; half traveling down.
- Velocity when it returns to the ground is the opposite of the velocity it was thrown upward with.
- Acceleration is –9.8 m/s2 everywhere!

Demonstration

- Object dropped from rest
- Object thrown up that falls.

Practice Problem

You drop a ball from rest off a 120 m high cliff. Assuming air resistance is negligible,

- how long is the ball in the air?
- what is the ball’s speed and velocity when it strikes the ground at the base of the cliff?
- what is the ball’s speed and velocity when it has fallen half the distance?
- sketch approximate x-vs-t, v-vs-t, a-vs-t graphs for this situation.

Announcements9/11/2014

Practice Problem

You throw a ball straight upward into the air with a velocity of 20.0 m/s, and you catch the ball some time later.

- How long is the ball in the air?
- How high does the ball go?
- What is the ball’s velocity when you catch it?
- Sketch approximate x-vs-t, v-vs-t, a-vs-t graphs for this situation.

Pretest Free Response

Case 1: Ball A is dropped from rest at the top of a cliff of height h as shown. Using g as the acceleration due to gravity, derive an expression for the time it will take for the ball to hit the ground.

A

vo

Pretest Free ResponseCase 2: Ball B is projected vertically upward from the foot of the cliff with an initial speed of vo. Derive an expression for the maximum height ymax reached by the ball.

B

Pretest Free Response

Case 3: Ball A is dropped from rest at the top of the cliff at exactly the same time Ball B is thrown vertically upward with speed vo from the foot of the cliff such that Ball B will collide with Ball A. Derive an expression for the amount of time that will elapse before they collide.

A

h

vo

B

Pretest Free Response

Case 4: Ball A is dropped from rest at the top of the cliff at exactly the same time Ball B is projected vertically upward with speed vo from the foot of the cliff directly beneath ball A. Derive an expression for how high above the ground they will collide.

A

h

vo

B

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