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One Dimensional Motion (Motion in a Straight Line)

Chapter 2

MOTION QUANTITIES

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Kinematics - Intro

Mechanics generally consists of two parts: Kinematics and Dynamics.

Mechanics

Dynamics Kinematics

Description of Motion Cause of Motion

Vectors and Scalars

Some quantities that we measure have only a magnitude (or size). We call these quantities “scalars”.

Some quantities that we measure have both magnitude and direction. We call these quantities “vectors”.

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Vectors and Scalars

We designate that a quantity is a vector by drawing an arrow “hat” over the variable that represents that quantity. Below are some examples:

Velocity = v

Acceleration = a

Force = F

Note: Because it is often difficult to place arrows over text, I will often use a BOLDED variable to signify a vector

Position and Displacement

Position is defined as the location of an object relative to some assigned origin or point of reference.

Variable: x or y

Units: m (meters)

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Position and Displacement

An easy way to represent position in one dimension is to use a number line, as shown below.

The origin or point of reference is designated by the “0”.

- + 0 1 2 3 4 5 -5 -4 -3 -2 -1

X (meters)

Sample 1 Below you see two “objects” A and B. a) What are their positions, relative to the origin? b) Does it make sense to say that B is at a greater position than A? Explain.

-5 -4 -3 -2 -1 X (meters)

- + 0 1 2 3 4 5

A B

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Position and Displacement

Displacement is defined as the change in position. Variable: Dx Equation: Dx = xf – x0 Distance is the scalar counterpart to displacement. It tells use how far an object has moved.

Sample 2

a) If an object moves from point A to point B, what is its displacement?

b) If an object moves from point B to point A, what is its displacement?

- + 0 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

A B

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Sample 3 An object moves from point B to point A and then to point C. a) What is the total distance that it traveled? b) What is the total displacement of the object?

- + 0 1 2 3 4 5 -5 -4 -3 -2 -1 X (meters)

A B C

Checkpoint 1 Here are three pairs of initial and final positions, respectively, along an x axis. Which pairs give a negative displacement? a) -3 m, +5 m b) -3 m, -7 m c) 7 m, -3 m

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Velocity ( ) is defined as the rate of change of position. A “rate” is how fast something changes over time. In the case of velocity, it is the position that changes. Mathematically, we can write this as: This equation states that the velocity is defined as the change in position divided by the time interval or change in time. Units: m/s

Velocity and Speed

t

xx

tt

xx

t

xv

f

f

f 0

0

0

D

D

v

Speed (v or s) is the scalar counterpart to velocity Speed is defined mathematically as: Speed is a scalar and, like distance, has no direction and cannot be negative.

Velocity and Speed

v d

tx

t

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We can characterize velocity in different ways, depending on how it is measured. Average velocity (which you are most familiar with) is defined as the change in position over the change in time over a “long” time. where Dt is a “long” time.

Average, Instantaneous and Uniform

vavg Dx

Dt =

In contrast, we can define an instantaneous velocity as a velocity “at an instant”. Mathematically, we would write this as: This means that the instantaneous velocity is just the average velocity as the time interval (Dt) approaches zero, i.e. at an instant. Instantaneous velocity is the derivative of position. Note: when we just say “velocity” we typically mean instantaneous velocity.

Average, Instantaneous and Uniform

dt

dx

t

xv

tinst

D

D

Dlim

0

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Sample 4 The position of a particle moving an an x axis is given by x = 7.8 + 9.2t -2.1t3, with x in meters and t in seconds. a) What is its velocity at t = 3.5 s? b) Is the velocity constant, or is it continuously changing?

Sample 5 You leave your house and drive a beat-up pickup truck along a straight road for 8.4 km at 70 km/h, at which point the truck runs out of gasoline and stops. Over the next 30 minutes, you walk another 2.0 km farther along the road to a gas station. a) What is your overall displacement from the beginning of your drive to

your arrival at the station? b) What is the time interval Δt from the beginning of your drive to your

arrival at the station? c) What is your average velocity from the beginning of your drive to your

arrival at the station? Find it both numerically and graphically. d) Suppose that to pump the gasoline, pay for it, and walk back to the

truck takes you another 45 minutes. What is your average speed from the beginning of your drive to your return to the truck with the gasoline?

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Checkpoint 2 In Sample 5, suppose that right after refueling the truck, exasperated, you drive back to your house. What is your velocity for your entire trip?

Checkpoint 3 The following equations give the position x(t) of a particle in four situations (in each equation, x is in meters, t is in seconds, and t > 0): 1) x = 3t -2 2) x = -4t2 – 2 3) x = 2/t2

4) x = -2

a) In which situation is the velocity v of the particle constant? b) In which is v in the negative x direction?

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Acceleration ( ) is defined as the rate of change of velocity. In other words, acceleration is how fast the velocity changes.

Mathematically, the definition of acceleration is Units: m/s2

Acceleration

a

vf vi - tf ti - aavg

Dv

Dt = =

vf vi - t

=

Acceleration

Instantaneous acceleration is defined as the derivative of velocity or the second derivative of position.

2

2

0lim

dt

xd

dt

dx

dt

d

dt

dv

t

va

t

D

D

D

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An Acceleration’s Sign

In common language, the sign of an acceleration has a non-scientific meaning: positive acceleration means that the speed of an object is increasing, and negative acceleration means that the speed is decreasing (the object is “decelerating”). However in physics, the sign of the acceleration indicates a direction, not whether an object’s speed is increasing or decreasing. The proper way to interpret the signs: If the signs of the velocity and acceleration or a particle are the same, the speed of the particle increases. If the signs are opposite, the speed decreases.

Checkpoint 4 A wombat moves along an x axis. What is the sign of its acceleration if it is moving a) in the positive direction with increasing speed b) in the positive direction with decreasing speed c) in the negative direction with increasing speed d) in the negative direction with decreasing speed

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Sample 6 A particle’s position on the x axis is given by x = 4 – 27t + t3, with x in meters and t in seconds. a) Find the particles velocity function v(t) and acceleration function a(t). b) Is there ever a time when v = 0? c) Describe the particle’s motion for t ≥ 0.

GRAPHING MOTION QUANTITIES

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Graphing Motion Quantities

You should be able to sketch graphs of any of the three motion quantities given a plot of any of the other quantities.

Derivatives [i.e. x(t) v(t)]

– Use slope

Integrals [i.e. v(t) x(t)]

– Use area

Graphing Velocity

Graphically, the velocity at any point is the slope of the curve of x(t) at that point.

1

X (m)

t (s)

2

3

4

5

0 1 2 3 4 5

m = 1 m/s 1

v (m/s)

t (s)

2

3

4

5

0 1 2 3 4 5

v = 1 m/s

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Sample 7 The graph below shows an x(t) plot for an elevator cab that is initially stationary, then moves upward (which we take to be the positive direction of x), and then stops. Plot v(t).

Graphing Acceleration

Graphically, the acceleration at any point is the slope of the curve of v(t) at that point.

1

X (m)

t (s)

2

3

4

5

0 1 2 3 4 5

t (s)

V (m/s)

1

2

3

4

5

0 1 2 3 4 5

v = 1 m/s

a (m/s2)

1

2

3

4

5

0 1 2 3 4 5

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Sample 8 For the scenario described in Sample 7, plot a(t).

Sample 9 Given the position function below, sketch a graph of the velocity function.

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X (m)

t (s)

10

15

20

25

0 1 2 3 4 5

30

35

v (m/s)

t (s)

0

0 1 2 3 4 5

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Sample 10 Given the velocity function below, sketch a graph of the position function.

x (m)

t (s) 0 1 2 3 4 5

0

-10

-5

0

5

10

15

20

5

v (m/s)

t (s)

10

15

20

25

0 1 2 3 4 5

30

35

Sample 11 In one test to study neck injury in rear-end collisions, a volunteer was strapped to a seat that was then moved abruptly to simulate a collision by a rear car moving at 10.5 km/h. The figure on the next slide gives the accelerations of the volunteer’s torso and head during the collision, which began at time t = 0. The torso acceleration was delayed by 40 ms because during that time interval the seat back had to compress against the volunteer. The head acceleration was delayed by an additional 70 ms. What was the torso speed when the head began to accelerate?

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Graphing Uniform Acceleration

What do you think a graph of uniform acceleration would look like on an acceleration vs. time graph? Assume the acceleration is a constant 2 m/s2. Then complete the velocity and position graphs.

X (m)

t (s) 0 1 2 3 4 5

t (s)

V (m/s)

0 1 2 3 4 5

a (m/s2)

1

2

3

4

5

0 1 2 3 4 5

1

2

3

4

5

1

2

3

4

5

t (s)

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CONSTANT ACCELERATION A Special Case

Derivations of Equations for Constant Accelerated Motion

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Equations for Motion with Constant Acceleration

Equation Missing Quantity

v = v0 + at x – x0

x – x0 = v0t + ½at2 v

v2 = v02 + 2a(x - x0) t

x – x0 = ½(v0 – v)t a

x – x0 v0

*Note: You may only use these equations when ACCELERATION IS CONSTANT

Checkpoint 5 The following equations give the position x(t) of a particle in four situations: 1) x = 3t -4 2) x = -5t3 + 4t2 + 6 3) x = 2/t2 4) x = 5t2 – 3

To which of these situations do the constant accelerated motion equations apply?

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Sample 12 Spotting a police car, you brake a Porsche from a speed of 100 km/h to a speed of 80.0 km/h during a displacement of 88.0 m, at a constant acceleration. a) What is that acceleration? b) How much time is required for the given decrease in speed?

FREE-FALL

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Free-Fall Acceleration

If the effects of air are eliminated, all objects accelerate downwards at a certain constant rate known and the free-fall acceleration.

The magnitude of free-fall acceleration is represented by g and has a value of 9.8 m/s2.

The constant accelerated motion equations apply to objects in free-fall.

Sample 13 On September 26, 1993, Dave Munday went over the Canadian edge of Niagara Falls in a steel ball equipped with an air hole and then fell 48 m to the water (and rocks). Assume his initial velocity was zero, and neglect the effect of the air on the ball during the fall. a) How long did Munday fall to reach the water surface? b) Munday could count off the three seconds of free fall but could

not see how far he had fallen with each count. Determine his position at each full second.

c) What was Munday’s velocity as he reached the water surface? d) What was Munday’s velocity at each count of one full second?

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Sample 14 In the provided figure, a pitcher tosses a baseball up along a y axis, with an initial speed of 12 m/s. a) How long does the ball take to

reach its maximum height? b) What is the ball’s maximum

height above its release point?

c) How long does the ball take to reach a point 5.0 m above its release point?

Checkpoint 6 In sample 13: a) What is the sign of the ball’s displacement for the ascent, from

the release point to the highest point?

b) What is it for the descent, from the highest point back to the release point?

c) What is the ball’s acceleration at its highest point?