Motion in One Motion in One DimensionDimension

Motion in One DimensionMotion in One Dimension

Movement along a straight-line path Linear motion

Convenient to specify motion along the x and y coordinate system

Motion in One DimensionMotion in One Dimension

Important to specify magnitude & direction of motion (Up or down; North, South, East, or West)

Coordinate (x and y) axesObjects right of origin on x axis = positive

Objects left of origin on x axis = negative

Objects above origin on y axis = positive

Objects below origin on y axis = negative

POSITION:Location of an object relative to an origin

Displacement versus DistanceDisplacement versus Distance

Displacement (x)Distance & direction

Measures net change in position

Displacement may not equal total distance traveled

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Δx = x f − x i

DistanceDistance

DistanceTotal path length traversed in moving from one location to anotherExample: Jimmy is driving to school but he forgets to pick up Johnny on the way…He now has to reverse his direction and drive back 2 miles

Total Distance Traveled = 12 miles

Total Displacement = 8 miles

+ 10 miles

- 2 miles

VectorVector

Quantity with both magnitude and direction

* Represented by arrows in diagrams

Displacement

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xo = initial position

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x = final position

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Δx = x − xo = displacement

VelocityVelocity

Average velocityAverage velocity is the displacement divided by the elapsed time.

timeElapsed

ntDisplaceme velocityAverage

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Average Velocity = v =Δx

Δt=x − x0

t − t0

Speed versus VelocitySpeed versus Velocity

Speed:Speed: Positive number, with units

Does not take into account direction

Speed is therefore a _ _ _ _ _ _ quantity?

Velocity (v):Velocity (v):Specifies both the magnitude (numerical value

how fast an object is moving) and also the direction in which an object is moving

Velocity is therefore a _ _ _ _ _ _ quantity?

Average VelocityAverage Velocity

The football player ran 50 m in 20 s. What is his average velocity?

v = 50 m / 20 s = 2.5 m/s

Velocity

The World’s Fastest Jet-Engine Car

Andy Green in the car ThrustSSC set a world record of 341.1 m/s (762.8 mph) in 1997. To establish such a record, the driver makes two runs through the course, one in each direction, to nullify wind effects. From the data, determine the average velocity for each run.

Velocity

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v =Δx

Δt=

+1609 m

4.740 s= +339.5m s

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v =Δx

Δt=

−1609 m

4.695 s= −342.7m s

(759 mph)

(766.4 mph)

AccelerationAcceleration

Acceleration (a)Acceleration (a) Change in velocity per time interval

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a =v − vot − to

=Δv

Δt

Animation of constant acceleration website

Acceleration

Acceleration and Acceleration and Decreasing Decreasing

VelocityVelocity

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a =v − vot − to

=13m s − 28m s

12 s − 9 s= −5.0m s2

Acceleration

AccelerationAcceleration

Acceleration:Vector quantity

Positive acceleration Velocity increasing

Negative acceleration Velocity decreasing

SI units for acceleration:meters/secondmeters/second2 2 m/s m/s22

Acceleration Animation Website

Animation - Direction of Acceleration and Velocity Website

Motion Equations for Motion Equations for Constant AccelerationConstant Acceleration

Constant Acceleration:Instantaneous & average accelerations are equal

1. Displacement x (meters)

2. Acceleration (constant) a (m/s2)

3. Final velocity v (m/s)

4. Initial velocity vo (m/s)

5. Elapsed time t (s)

Variables

Motion Equations for Motion Equations for Constant AccelerationConstant Acceleration

tvvx o 21

221 attvx o

atvv o

axvv o 222

Solving Problems

Draw a depiction of the situation

Utilize x and y coordinate axes with +/- directions

Write down known variables

Select appropriate equation

Complete calculation**UNITS!

Reasonable result?

Example: CExample: Catapulting a Jetatapulting a Jet

** Find its displacement.

sm0ov

??x

2sm31a

sm62v

a

vvvvtvvx o

oo

2

121

a

vvx o

2

22

m 62

sm312

sm0sm62

2 2

2222

a

vvx o

An Accelerating SpacecraftAn Accelerating Spacecraft

A spacecraft is traveling with a velocity of +3250 m/s. Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s2. What is the velocity of the spacecraft when the displacement of the craft is +215 km, relative to the point where the retrorockets began firing?

x a v vo t

+215000 m -10.0 m/s2 ? +3250 m/s

axvv o 222

x a v vo t

+215000 m

-10.0 m/s2

? +3250 m/s

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v = vo2 + 2ax

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v = (+3250m /s)2 + 2(−10.0m /s2)(+215000m)

v = 2503 m/s

Falling ObjectsFalling Objects

Galileo Galilei’sGalileo Galilei’s Contribution In the absence of air resistance, all objects

on Earth fall with the same constant acceleration. Acceleration due to gravity (g) = 9.8m/s2

Free-fallFree-fall

Any freely falling object being acted upon solely by the force of gravity

Ignore air resistance

Rate of acceleration due Earth’s gravity

g = 9.8 m/s2

Vector Direction is towards the center of the Earth

Free FallFree Fall

Object does not have to be falling to be in free fall Example - Throwing a ball upward Motion is

still considered to be free fall, since it is moving under the influence of gravity

Constant AccelerationConstant AccelerationEquationsEquations

tvvx o 21

221 attvx o

atvv o

axvv o 222

Acceleration due to Acceleration due to Gravity EquationsGravity Equations

gtvv o

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y = 12 vo + v( ) t

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v 2 = vo2 + 2gy

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y = vot +12 gt

2

A Falling StoneA Falling Stone

A stone is dropped from the top of a tall building. After 3.00s of free fall, what is the displacement y of the stone?

y a v vo t

? - 9.80 m/s2 0 m/s 3.00 s

y a v vo t

? - 9.80 m/s2 0 m/s 3.00 s

m 1.44

s 00.3sm80.9s 00.3sm0 2221

221

attvy o

How High Does it Go?How High Does it Go?

The referee tosses the coin upwith an initial speed of 5.00m/s.In the absence of air resistance,how high does the coin go aboveits point of release?

y a v vo t

? - 9.80 m/s2 0 m/s + 5.00 m/s

y a v vo t

? -9.80 m/s2 0 m/s +5.00 m/s

ayvv o 222 a

vvy o

2

22

m 28.1

sm80.92

sm00.5sm0

2 2

2222

a

vvy o