Chapter 2

Motion in One Dimension(Kinematics)

2.1 Displacement and Velocity

• Distance is a measure of the total motion of an object (how far it has traveled)

• Displacement is the net distance and direction an object has traveled (straight line distance and direction from starting to ending point)

o What’s your distance when you run two laps around a 400 m track?

o What is your displacement?o How are they different?

What is the gecko’s displacement?

• ∆x = xf - xi

• ∆x = 85 cm – 25 cm• ∆x = 60 cm

Displacement continued• Displacement is not always equal to the

distance traveled (if you move back and forth)• Displacement can be positive or negative.

Speed• We use the variable speed to describe how quickly

something moves.• Saying a race car, runner, or plane is “fast” is not enough to

accurately describe its speed scientifically.

Speed

v = d t

Distance traveled (m)

Time taken (sec)

Speed (m/sec)

Calculate speed

• A bird flies 50 meters in 7.5 seconds. • Calculate the speed of the bird in m/sec.

Calculate distance

• How far do you go if you drive for 2 hours at a speed of 100 kilometers per hour?

Calculate time and distance

• A space shuttle is traveling at a speed of 7,700 m/sec.

• How far does the shuttle travel in kilometers in one hour?

• At an altitude of 300 kilometers, the circumference of the shuttle’s orbit is 42 million meters.

• How long does it take the shuttle to go around the Earth one time?

Calculating Speed

• Why the letter v is used for speed...o We use the letter v to represent speed in a formula. o We already used the letter s for seconds

• Speed is a single measurement that tells how fast you are going, like 100 km/h.

• Velocity means you know both your speed and your direction.

Instantaneous Speed and Velocity

• Instantaneous speed is distance ÷ time interval o when the time interval is made very small

t

xvinst

Average Speed• Average Speed is the TOTAL DISTANCE divided by the

TOTAL TIME

o When is Average Speed = Instantaneous speed?

tot

totav T

Dv

Velocity• Average velocity is displacement divided by

the time interval

2.2 Acceleration• Acceleration measures the rate of change of

velocity• When an object speeds up or slows down it is

accelerating.• When an object is speeding up its acceleration

is positive and when it is slowing down its acceleration is negative

• Acceleration movie

Constant Acceleration

• Constant acceleration is sometimes called uniform acceleration.

• A ball rolling down a straight ramp has constant acceleration.

constant acceleration increasing speed

Acceleration and Speed

• An object can have acceleration, but no speed. • Consider a ball rolling up a ramp.• As the ball slows down, eventually its speed

becomes zero.

constant negative acceleration decreasing speed

Speed vs Velocity• We frequently use the terms speed and

velocity interchangeably• But remember that velocity is speed in a

particular direction• So speed is always a positive number • Whereas velocity can be a negative number if

you are going backwards.

Slope and Acceleration

• Use slope to recognize when there is acceleration in speed vs. time graphs. o Level sections (A) on the graph

show an acceleration of zero.

o The highest acceleration (B) is the steepest slope on the graph.

o Sections that slope down (C) show negative acceleration (slowing down).

Slope of a graph

• The slope of a graph is equal to the ratio of rise to run.

• The rise is the amount the speed changes.

• The run is the amount the time changes.

Acceleration and slope• Acceleration is the change in speed over the change in time.• The slope of a speed vs time graph is acceleration.

Calculating Acceleration

• The formula for acceleration can be expressed several ways

But we often write it as:

Acceleration units• Acceleration is a unit of distance divided by a

unit of time squared.• In physics we generally use the unit: m/s2

(meters per second per second)

• A biker goes from a speed of 0.0 m/s to a final speed of 25.0 m/s in 10 seconds. What is the acceleration of the bicycle?

Kinematics Equations• You should now know how to calculate

average velocity and average acceleration using the equations for velocity and acceleration.

• Next we will combine them so that you can solve any problem related to objects that are moving at a constant velocity or accelerating.

• We need 3 more equations to do this.

Finding displacement (∆x) when

you know initial and final velocity

and time

• The first equation allows you to find displacement when you know the initial and final velocity of an object and the time that it accelerated.

Question: A sports car accelerates from rest to a velocity of 25 m/s in 5.0 seconds.

How far did it travel?

Finding displacement (∆x) when you know initial velocity, acceleration, and

time.

• The second equation allows you to find displacement when you know initial velocity, acceleration, and time (but not final velocity).

• You used this equation to determine how high your tennis ball went.

Question: A car traveling at 30 m/s brakes for 3.0 seconds at an acceleration of -2.0 m/s2. How far did it travel during this time?

• )

Finding final velocity when you know initial velocity, acceleration, and time• The third equation is used to final velocity

when you know initial velocity, acceleration, and time

Question: A jet plane accelerates from rest down a runway for 12 seconds at an acceleration of 15 m/s2. What is it’s final velocity?

Using the Equations of Kinematics

• Choose a positive direction• Draw a picture• Make a chart of initial conditions• Choose an equation to solve for the

unknown• Solve and check answer for reasonability

Kinematics Example #1 A car accelerates from a stop light for

20.0 s with an acceleration of 5.0 mph/s2. Find its velocity at the end of this time. How far has it traveled?

Kinematics Example #1

V = 100 mphX = 0.278 miles

Kinematics Example #2

A 747 airliner needs to achieve a speed of 360 km/h to take off. If acceleration is uniform and the runway is 1.8 km long, what acceleration is needed?

Kinematics Example #2

10 km/h/s or 2.78 m/s2

Kinematics Example #3

A penny is dropped from a building rooftop and takes 6.0 s to hit the ground. What velocity does it hit the ground with? How tall is the building in meters? in feet? How would this change if you accounted for the speed of sound?

Kinematics Example #3

H=176m = 579ftV=58.8 m/s

Example #4 A High Speed Chase

• In a high speed chase, a police car takes off after a getaway car going at its maximum speed of 90 mph. The police car accelerates at a rate of 10 mph/s to a maximum speed of 120 mph. How long does it take to catch the bad guys?

Free Fall• An object is said to be in “free

fall” if it experiences only a force of gravity.

oWe abbreviate the accel. of gravity as “g”og=9.81m/s2 near the Earth’s surface

What goes up…• A ball is tossed in the air..oWhen is its velocity 0?oWhen is its acceleration positive?

Negative?

Kinematics Example # 4• A ball is thrown straight up in the air with an

initial velocity of 25.0 m/s. How long will it take for the ball to reach its starting point? What will the ball’s velocity be when it returns to its starting point?

Kinematics Example # 6

t = 5.08 svf = 25.16 m/s

The End…