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Chapter 2: 1-D Kinematics

Brent Royuk Phys-111

Concordia University

Displacement •  Levels of Formalism •  The Cartesian axis

–  One dimension: the number line

•  Mathematical definition of displacement:

–  delta means change

•  What’s the difference between distance and displacement?

•  Example: If you travel from xi = -5 m to xf = -1 m what is your displacement?

Δx = xf− x

o

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Vectors and Scalars •  Vectors are numbers with directions.

– Often symbolized by arrows

• Scalars have magnitudes but no direction.

• Chapter 2 motion is all one-dimensional. – Most important: +/- refer to

direction: left/right or up/down usually

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Vectors and Scalars

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Average Speed and Velocity •  Speed: how fast an object’s position is

changing •  To measure speed, measure distance

and time

If it’s not average speed, what kind of speed is it?

– MKS unit: m/s – How does a m/s compare with a mph?

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average speed =distance

elapsed time

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Average Speed and Velocity •  Vector speed: velocity (more later)

• A note on notation: Relax. – When speed/velocity is uniform,

problems are doable.

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average velocity = displacementtotal travel time

vav=ΔxΔt

=xf− x

o

tf− t

o

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v = x - xo

t − to

, x = xo + vt and x = vt

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Example

•  Take a canoe trip down-river 2 miles at 2 mph. Then return upriver 2 miles at 1 mph. What is your average speed? What is your average velocity?

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Motion Graphs

•  The abscissa and the ordinate •  Uniform v, positive and negative •  The slope triangle

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Motion Graphs

•  The abscissa and the ordinate •  Uniform v, positive and negative •  The slope triangle

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Instantaneous Velocity

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Examples •  A fast sprinter can run the 100-m dash in

10.0 s, and a good marathoner can run the marathon (26mi, 385 yd) in 2hr 10’. a) what are their average speeds? Calculate to three sig-figs. b) if the sprinter could maintain sprint speed during the marathon, how long would it take?

•  A bike travels around a circular track of diameter 500.0 m at a constant speed in 2 min, 10 sec (130.0 s). What is the speed of the bike? Is the velocity constant?

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Examples •  A jet is flying 35m above level ground at a

speed of 1300 km/h. Suddenly the plane encounters terrain that gently slopes upward at 4.3o, an amount difficult to detect. How much time does the pilot have to make a correction before flying into the ground?

•  Two trains, each having a speed of 30 km/h, are headed toward each other on the same track. A bird that can fly 60 km/h flies off the front of one train when they are 60 km apart and flies back and forth between the trains until they crash. How far does the bird fly?

•  Assignment: Homework 2A

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Ranking Task

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Ranking Task

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Acceleration

•  – The MKS acceleration unit – Everyday accelerations

•  What is deceleration? •  Change in direction but not magnitude

of velocity is still an acceleration– e.g. car going around curve at constant speed

•  Jerk, snap, crackle & pop

aav=ΔvΔt

=vf−v

o

tf− t

o

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Graphing Acceleration •  V-t graphs •  What is the slope? •  The area under the curve •  Average vs. instantaneous

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Constant Acceleration • When acceleration is held

constant, we can write

• Compare with y = mx + b

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a = v − vo

t and v = vo + at

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Constant Acceleration • How can you figure out how far an

object travels while accelerating?

•  In practice, we can usually let xo=0 without losing generality, so

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vav = 12

vo + v( ) , so x = xo +12

vo + v( )t

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x =vo + v( )

2t

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Constant Acceleration •  A car accelerates from 12 m/s to 18 m/s over

a time of 6.0 s. How far did it travel during this time?

•  A particle had a velocity of 18 m/s and 2.4 s later its velocity was 30 m/s in the opposite direction. What was its average acceleration during this time? How far did it travel?

•  Assume the brakes on your car are capable of creating a deceleration of 5.2 m/s2. If you are going 85 mph and suddenly see a state trooper, how long will it take to decelerate to 55 mph? How far do you travel during this time?

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Constant Acceleration •  Start with •  Eliminate v to get

– Note the t2 dependence – See it on an x-t graph

•  Eliminate t to get

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x = v +vo

2t and v = vo + at

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x = vot + 12

at 2

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v2 = vo2 + 2ax

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The Complete Equation List

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v = vo + at

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x = vot + 12

at 2

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v2 = vo2 + 2ax

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x =vo + v( )

2t

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x = vt

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Problem-Solving Basics 1.  Examine the situation to determine which physical

principles are involved. 2.  Make a list of what is given or can be inferred from

the problem as stated (identify the knowns). 3.  Identify exactly what needs to be determined in the

problem (identify the unknowns). 4.  Find an equation or set of equations that can help

you solve the problem. 5.  Substitute the knowns along with their units into

the appropriate equation and obtain numerical solutions complete with units.

6.  Check the answer to see if it is reasonable. Does it make sense?

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Examples •  If my motorcycle could maintain its 0-60 acceleration

for 15 seconds, how far would it have traveled? How fast would it be going?

•  If you brake your Porsche from 52 m/s to 24 m/s over a distance of 120 m, what was your acceleration?

•  The driver of a pickup truck going 1.00x102 km/h applies the brakes, giving the truck a uniform deceleration of 6.50 m/s2 until it slows to 5.00 x 101 km/h. a) How far did the truck travel while it was braking? b) How much time has elapsed?

•  A subway train accelerates from rest at one station at a rate of 1.2 m/s2 for half the distance to the next station, then decelerates at this same rate for the final half. If the stations are 1100m apart, find a) the time of travel between stations and b) the maximum speed of the train.

•  Assignment: Homework 2B 25

Falling Things •  Galileo and his

famous experiment

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Falling Things •  Falling objects accelerate at the same rate, g

= 9.8 m/s2 – The acceleration is uniform

• This is not obvious; Aristotle thought falling objects acquired some characteristic falling speed.

• He was thinking of terminal velocity, perhaps?

• Galileo: a rock falling 2 m drives a stake much more than a rock falling 10 cm.

•  Linear accelerations (or any type of acceleration) can be expressed in g’s: fighter pilots, etc.

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Falling Things •  g is altitude and latitude-dependent

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Falling Things •  g is altitude and latitude-dependent

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Falling Things •  g is depth-dependent

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Falling Things •  ONLY IN VACUUM

–  Astronaut David Scott, 1971: hammer and feather

–  video –  Painting by astronaut Alan

Bean •  g on the Moon

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Apollo 15

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Feather Drop

Ranking Task

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Falling Problems •  Equations

a = -g can be written explicitly into kinematics equations •  This assumes + is up •  also replace x with y

•  Examples – Make a table with columns for t,y,v,a. Fill

in values for t = 0,1,2,3,4 – A wrench is dropped down a 45 m elevator

shaft. How fast does it hit? How long does it take to hit?

– A cliff-diver steps off a cliff and falls 3.1s before hitting water. How high is the cliff? How fast is the diver falling when he hits? 38

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Falling Problems •  A stone is thrown downward from a bridge with a speed

of 14.7m/s. If the stone hits the water in 2.00 s, what is the height of the bridge? –  What if it was 14.7m/s up? What does this answer

mean? •  A pitcher throws a baseball straight up, with vo = 25 m/

s. a) How long does it take to reach its highest point? b) How high does the ball rise above its release point? c) How long will it take for the ball to reach a point 25m above its release point?

•  A rock is dropped from a 100m cliff. How long does it take to fall the first 50m and the last 50m? Apply to “hang time.”

•  Reaction time and the dollar trick. •  Assignment: Homework 2C

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