KinematicsKinematics

Relative MotionRelative Motion

• Motion is relative

• The same event, viewed from two different points of view, can yield two different measurements

Quantities of InterestQuantities of Interest

• Position: where you are relative to a specific origin

• Elapsed Time: measurement of a clock

• Speed/Velocity

• Acceleration

Distance/DisplacementDistance/Displacement

• Distance: your entire trip

• Displacement: difference between initial and final positions

• If you backtrack, or travel in multiple directions, these two numbers will be different

Speed/VelocitySpeed/Velocity

• Speed depends upon distanced traveled

Speed = distance traveled/time

• Velocity depends upon displacement

Velocity = displacement/time

• Which has direction?

• Assuming humans originated in Africa and migrated to other parts of the world, some time would be required for this to occur. At the modest rate of 1km/year, how many centuries would it take humans to migrate from Africa to China, some 10,000km away?

• Is it possible for an object to change velocities while holding a constant speed?

• I travel 20 miles N in 30 minutes, then 60 miles south in 90 minutes. What is my average velocity?

• .33 miles/min or 20 miles/hour

• South

• On a car trip, I travel at 60 miles/hour for 2 hours, stop and rest for 30 minutes, then travel at 70 miles/hour for 4 hours. How far do I travel? What is my average speed? (if you’re feeling ambitious, draw a position/time graph for this trip)

• 400 miles• 61.5 miles/hour

Turnpike TicketsTurnpike Tickets

• Regardless of how sneaky you might be, it’s possible to get caught speeding on turnpikes where you pick up a ticket at the entrance and drop off the ticket at the exit

• How?

Average vs. InstantaneousAverage vs. Instantaneous

• Average quantities corresponding to lengths of time

• Instantaneous quantities correspond to instants in time

• Mathematically, we’re looking at the limit of a function as t approaches 0

Instantaneous VelocityInstantaneous Velocity

• An object’s velocity at a particular instant in time

• We can figure out its instantaneous velocity by looking at a position/time graph

• As we compute position and time differences over shorter and shorter intervals, we approach our instantaneous value (see Walker, p23)

• As our time interval decreases, our slope (velocity) approaches a constant value

• Why?

• All functions, even curves, are linear on small enough scales

Speed? Speed?

• The magnitude of our instantaneous velocity is instantaneous speed

• We’re all familiar with a device that measures instantaneous speed

• What is it?

Graphical Views of MotionGraphical Views of Motion

• Stationary (on x and v graphs)

• Constant speed (on both)

• Speeding up (on both)

• Slowing down (on both)

• Moving backwards (on both)

Acceleration and FreefallAcceleration and Freefall

Questions of SignificanceQuestions of Significance

• If you drop a penny off the Empire state building, how fast is moving when it hits the ground?

• If you launch an object into the air, how long does it take to hit the ground?

• How can you calculate a snow-boarder’s hang time? LeBron James’ hangtime?

• What is the minimum length necessary for an airport runway?

Velocity/Time GraphsVelocity/Time Graphs

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12 14

Time (s)

Vel

oci

ty (

m/s

)

SlopesSlopes

• On a position/time graph, the slope represents the object’s velocity

• How about the slope on a velocity/time graph?

• Slope = rise/run

• Slope = v/t

Slope Units?Slope Units?

• Slope = v/t

• Slope = m/s/s = m/s2

• What does the slope physically represent?

• The rate of change in velocity

• We call this quantity, acceleration

• Like velocity, it is a vector quantity

The Meaning of Acceleration UnitsThe Meaning of Acceleration Units

• m/s/s; what does this mean?

• Let’s think about gravity, which accelerates objects at about 10m/s/s

• If you drop an object from rest, how fast will it fall?

• After the first second, 10m/s; the second 20 m/s; the 3rd, 30m/s

Acceleration ValuesAcceleration Values

• Acceleration due to gravity: 9.8 m/s/s

• Honda Civic: 3.0 m/s/s

• Jumbo Jet: 2.5 m/s/s

• Space Shuttle: 20 m/s/s

Negative AccelerationNegative Acceleration

• Acceleration is a vector, which means it has a direction

• If I travel in the + direction, but my acceleration is in the negative direction, what happens?

• Ex: braking, throwing keys in the air

Displacement?Displacement?

• We know the relationship between acceleration and velocity

• How does displacement fit into the picture?

• Ex: How far does a car travel as it accelerates from 0 to 60mph?

• Imagine a rock, thrown downwards off a cliff at a speed of 30m/s

• I start my clock when the rock is 2m below the edge of the cliff

• Fill in the following table of information concerning this rock:

Time

(s)

Inst.

Velocity

(m/s)

Average Velocity

(s)

x

(m)

Position

(m)

0

1

2

3

4

5

Time

(s)

Inst.

Velocity

(m/s)

Average Velocity

(m/s)

x

(m)

Position

(m)

0 30 ____ ___ 2

1 40 35 35 37

2 50 45 45 82

3 60 55 55 137

4 70 65 65 202

5 80 75 75 277

• Graph position data vs. time data for this fall period and fit with the appropriate function

• How does position depend upon time?

0

50

100

150

200

250

300

0 1 2 3 4 5 6

Time (s)

Po

siti

on

(m

)

• Equation of the fit

y(t) = 5t2 + 30t + 2

• What do the fit coefficients physically represent?

• y = final position

• 2 = initial position (meters)

• 30 = initial velocity (m/s)

• 5 = ½ acceleration (m/s2)

• yf = 1/2at2 + vit + yi

• yf – yi = 1/2at2 + vit

• y = vit + 1/2at2

Applying DVATsApplying DVATs

• How far does a Porsch travel if it accelerates from 0 – 30 m/s (60 mph) over a time interval of 6s?

The Deadly PennyThe Deadly Penny

• Will a penny, dropped from the Empire State building, kill someone on the ground below?

How to Solve Physics ProblemsHow to Solve Physics Problems

• 1. Draw a picture (with initial and final)• 2. Think about the following questions

• What am I trying to find? • What do I know?• What do I need to know?

• 3. Think about the physics at play• 4. Find the appropriate mathematics• 5. Solve it (in symbols first!)• 6. Does your answer make sense?

• What are we trying to find?• Final velocity of a penny

• What do we know?• Initial velocity, acceleration due to gravity

• What do we need to know?• Height of building, fatal drop speed

Useful informationUseful information

• Empire State Building Height• Height = 381 m

• Velocity of a bullet• Velocity = 300 – 400 m/s

DVAT EquationsDVAT Equations

• 1: vf = vi + at

• 2: x = vit + ½ at2

• 3: x = ½(vf+vi) t

• 4: 2ax = vf2-vi

2

The Catch 22The Catch 22

• The underlying assumption of these equations is constant acceleration

• If we don’t have constant acceleration, we can’t use them…

Situation 2: Human AccelerationSituation 2: Human Acceleration

• Asafa Powell, the world’s fastest human, accelerates at a rate of +5m/s/s over a distance of 10m

• Assuming he starts from rest, what is his final velocity?

Situation 3: Braking DistanceSituation 3: Braking Distance

• According to the Highway Patrol, it takes about 75m to slow down from 70mph (35m/s) on dry road conditions

• What is your braking acceleration?

• How long does it take to stop?

• If you are traveling at 20m/s, how much braking distance do you need?

Impact SpeedImpact Speed

• Let’s revisit the previous situation

• Say you only have 20m before you hit the car in front of you (initial velocity = 35m/s)

• At what speed will you hit the car?

• How does human reaction time change these numbers?

Two Cars in Motion?Two Cars in Motion?

• Two cars, separated by 30m, both slam on their brakes at the same time

• Car 1, initially traveling at 35m/s, has an acceleration of -4.0m/s/s

• Car 2, initially traveling at 20m/s, has an acceleration of -8.0m/s/s

• At what speed will car 1 strike car 2?

Free FallFree Fall

• Neil Armstrong video on youtube

• http://www.youtube.com/watch?v=5C5_dOEyAfk

Problem TypesProblem Types

• How high? (max height on a toss)

• How long? (drop/hang time)

• How fast? (final velocity)

• Lebron James takes off from the ground with a vertical velocity of +5.5m/s

• How long is he in the air?

• How high does he go?

AssumptionsAssumptions

• vf = 0 at apex

• vi = -vf (assuming starting and ending heights are the same)

• Acceleration = -10m/s/s

• Time up = Time down

• In a fit of rage, a student leans out his 3 story window (10m in height) and throws his physics textbook straight up with an initial velocity of 8m/s. Assuming this student’s rage has sucked in all the surrounding air:

• How long will it take for the book to hit the ground below?

• At what speed will it strike the ground?

A person standing by the edge of a cliff A person standing by the edge of a cliff throws one ball straight up and another throws one ball straight up and another straight down at the same initial speed. straight down at the same initial speed.

Neglecting air, the ball to hit the ground with Neglecting air, the ball to hit the ground with the greater speed is the one initially thrown:the greater speed is the one initially thrown:

• 1. upward

• 2. downward

• 3. neither—they both hit the ground at the same speed

The graph on the following page The graph on the following page maps the position of two trains, maps the position of two trains,

A and B. Which statement A and B. Which statement below is true? below is true?

• 1. At time tB, both trains have the same velocity

• 2. Both trains have the same speed at all times• 3. Both trains have the same velocity at some time

before tB

• 4. Somewhere on the graph, both trains have the same acceleration

The position/time graph on the The position/time graph on the next page maps the motion of 4 next page maps the motion of 4 objects. Answer the following objects. Answer the following

questions related to these questions related to these object’s motion object’s motion

Rank the object’s average Rank the object’s average velocities in increasing ordervelocities in increasing order

• 1. A, B, C, D 2. B, A, D, C

• 3. A, C, D, B 4. B, D, C, A

• 5. C, A, D, B

Which object has the highest Which object has the highest instantaneous velocity (at any point instantaneous velocity (at any point

during the time interval?)during the time interval?)

• 1. A 2. B

• 3. C 4. D

Two identical objects are dropped from different Two identical objects are dropped from different heights. Object 1, dropped from height h, heights. Object 1, dropped from height h,

reaches a speed v when it hits the ground. reaches a speed v when it hits the ground. Assuming object 2 is dropped from height 2h, Assuming object 2 is dropped from height 2h,

how fast is it traveling when it hits the ground?how fast is it traveling when it hits the ground?

• 1. v/2 2. √2v

• 3. 2v 4. 4v

If I throw an object up in the air at If I throw an object up in the air at speed v, it rises 6m above my speed v, it rises 6m above my

hand. If I throw that same object hand. If I throw that same object on the moon (a = 1.6 m/s/s) with on the moon (a = 1.6 m/s/s) with speed v, how high will it travel? speed v, how high will it travel?

A speeding car traveling at a A speeding car traveling at a constant 30m/s passes a cop, constant 30m/s passes a cop,

initially at rest. If the cop initially at rest. If the cop accelerates uniformly at accelerates uniformly at

+4m/s/s, how long does it take +4m/s/s, how long does it take him to catch the speeder? him to catch the speeder?

A stone is dropped from the roof of a A stone is dropped from the roof of a tall building. 1.0s later, a second stone tall building. 1.0s later, a second stone

is dropped. How far apart are the is dropped. How far apart are the stones when the second one has stones when the second one has

reached a speed of 15.0m/sreached a speed of 15.0m/s