chapter 2 scalars kinematics: description of motion...

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Chapter 2 Kinematics: Description of

Motion

© 2010 Pearson Education, Inc. 1

• A scalar quantity is a quantity that has magnitude only and has no direction in space

Scalars

Examples of Scalar Quantities: }  Length }  Area }  Volume }  Time }  Mass

• A vector quantity is a quantity that has both magnitude and a direction in space

Vectors

Examples of Vector Quantities: }  Displacement }  Velocity }  Acceleration }  Force

2.1 Distance and Speed: Scalar Quantities

Distance is the path length traveled from one location to another. It will vary depending on the path.

Distance is a scalar quantity—it is described only by a magnitude.



Average speed is the distance traveled divided by the elapsed time:



Since distance is a scalar, speed is also a scalar (as is time).

Instantaneous speed is the speed measured over a very short time span.

© 2010 Pearson Education, Inc.

You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, have you and your dog traveled the same distance?

a) yes

b) no

Question 2.1 Walking the Dog

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2.2 One-Dimensional Displacement and Velocity: Vector Quantities

A vector has both magnitude and direction. Manipulating vectors means defining a coordinate system, as shown in the diagrams to the left.



Displacement is a vector that points from the initial position to the final position of an object.



Note that an object’s position coordinate may be negative, while its velocity may be positive; the two are independent.



For motion in a straight line with no reversals, the average speed and the average velocity are the same.

Otherwise, they are not; indeed, the average velocity of a round trip is zero, as the total displacement is zero!



Different ways of visualizing uniform velocity:



This object’s velocity is not uniform. Does it ever change direction, or is it just slowing down and speeding up?


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You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement?

a) yes

b) no

Question 2.2 Walking the Dog

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Question 2.2 Displacement

Does the displacement of an object

depend on the specific location of

the origin of the coordinate system?

a) yes

b) no

c) it depends on the coordinate system


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Question 2.2 Velocity in One Dimension

If the average velocity is non-zero over

some time interval, does this mean that

the instantaneous velocity is never zero

during the same interval?

a) yes

b) no

c) it depends

2.3 Acceleration

Acceleration is the rate at which velocity changes.


2.3 Acceleration Acceleration means that the speed of an object is changing, or its direction is, or both.


2.3 Acceleration

Acceleration may result in an object either speeding up or slowing down (or simply changing its direction).


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2.3 Acceleration If the acceleration is constant, we can find the velocity as a function of time:

© 2010 Pearson Education, Inc. 19 20

If the position of a car is

zero, does its speed have to

be zero?

a) yes

b) no

c) it depends on the position

Question 2.3 Position and Speed

2.4 Kinematic Equations (Constant Acceleration) From previous sections:

© 2010 Pearson Education, Inc. 21 22

You drive for 30 minutes at 30 mi/hr and then for another 30 minutes at 50 mi/hr. What is your average speed for the whole trip?

a) more than 40 mi/hr

b) equal to 40 mi/hr

c) less than 40 mi/hr

Question 2.4 Cruising Along I

2.4 Kinematic Equations (Constant Acceleration)

Substitution gives:

and:


2.4 Kinematic Equations (Constant Acceleration)

These are all the equations we have derived for constant acceleration. The correct equation for a problem should be selected considering the information given and the desired result.


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Vector Diagrams

•  Vector diagrams are diagrams which use vector arrows to depict the direction and relative magnitude of a vector quantity.

•  Vector diagrams can be used to describe the velocity of a moving object during its motion.

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Vector Diagrams

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Describing Motion with Position vs. Time

Graphs The Meaning of Shape

for a p-t Graph

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Constant Velocity •  To begin, consider a

car moving with a constant, rightward (+) velocity - say of +10 m/s.

•  Note that a motion described as a constant, positive velocity results in a line of constant and positive slope when plotted as a position-time graph.

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Changing Velocity

•  Now consider a car moving with a rightward (+), changing velocity (acceleration) - that is, a car that is moving rightward but speeding up or accelerating

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The position vs. time graphs for the two types of motion - constant velocity and changing velocity (acceleration) - are depicted as follows.

Constant Velocity Positive Velocity

Positive Velocity Changing Velocity (acceleration)

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Importance of slope

•  If the velocity is constant, then the slope is constant (i.e., a straight line).

•  If the velocity is changing, then the slope is changing (i.e., a curved line).

•  If the velocity is positive, then the slope is positive (i.e., moving upwards and to the right).

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Slope of p vs t

Slow, Rightward (+) Fast, Rightward (+) Constant Velocity Constant Velocity

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Slope

Slow, Leftward (-) Fast, Leftward (-) Constant Velocity Constant Velocity

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Meaning of slope

Negative (-) Velocity Leftward (-) Slow to Fast Fast to Slow

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Determining the Slope on a p-t Graph

•  In this part of the lesson, we will examine how the actual slope value of any straight line on a graph is the velocity of the object.

•  Consider a car moving with a constant velocity of +10 m/s for 5 seconds. The next diagram depicts such a motion.

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•  The slope of the line is +10 meter/1 second. It is obvious that in this case the slope of the line (10 m/s) is the same

as the velocity of the car

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•  Now consider a car moving at a constant velocity of +5 m/s for 5 seconds, abruptly stopping, and then remaining at rest (v = 0 m/s) for 5 seconds.

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Determining the slope

•  The line is sloping upwards to the right. But mathematically, by how much does it slope upwards per 1 second along the horizontal (time) axis? To answer this question we must use the slope equation.

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Check your understanding

•  Answer: -3.0 m/s

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The Meaning of Shape for a v-t Graph

•  Consider a car moving with a constant, rightward (+) velocity - say of +10 m/s. As learned in an earlier lesson, a car moving with a constant velocity is a car with zero acceleration.

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•  Note that a motion described as a constant, positive velocity results in a line of zero slope (a horizontal line has zero slope) when plotted as a velocity-time graph. Furthermore, only positive velocity values are plotted, corresponding to a motion with positive velocity.

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•  Now consider a car moving with a rightward (+), changing velocity - that is, a car that is moving rightward but speeding up or accelerating.

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•  The velocity vs. time graphs for the two types of motion - constant velocity and changing velocity (acceleration) - can be summarized as follows

Positive Velocity Zero Acceleration

Positive Velocity Positive Acceleration

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Notice that the slope of a velocity-time graph represents the acceleration of the object

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•  Now how can one tell if the object is speeding up or slowing down? Speeding up means that the magnitude (the value) of the velocity is getting large

You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation?

a) the separation increases as they fall

b) the separation stays constant at 4 m

c) the separation decreases as they fall

d) it is impossible to answer without more information

Question 2.5 Throwing Rocks I

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You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their velocities?

a) both increase at the same rate

b) the velocity of the first rock increases faster than the velocity of the second

c) the velocity of the second rock increases faster than the velocity of the first

d) both velocities stay constant

Question 2.5 Throwing Rocks II

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