2.1 Displacement and Distance

Position, displacement, distance, and path length

• These quantities describe the location and motion of an object.
• Position is an object's location with respect to a particular coordinate system.
• Displacement is a vector that starts from an object's initial position and ends at its final position.
• Magnitude of the Displacement is the magnitude (length) of the displacement vector.
• Distance Traveled is how far the object moved as it traveled from its initial position to its final position.

Imagine laying a string along the path the object took. The length of the string is the distance traveled.

Example: A car backs up (moving in the negative direction) toward the origin of the coordinate system at x = 0. The car stops and then moves in the positive x-direction to its final position x_f.

Example: A car backs up (moving in the negative direction) toward the origin of the coordinate system at x = 0. The car stops and then moves in the positive x-direction to its final position x_f.

Example: A car backs up (moving in the negative direction) toward the origin of the coordinate system at x = 0. The car stops and then moves in the positive x-direction to its final position x_f.

2.3 Time

Time and time interval

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• The time t is a clock reading.
• The time interval (t₂ − t₁) or Δt is a difference in clock readings. (The symbol Δ (delta) represents "change in" and is the final value minus the initial value.)
• These are both scalar quantities.
• The SI units for both quantities are seconds (s).

A stopwatch is used to measure a time interval.

2.3 Average Velocity

• If an object is moving steadily, with a constant velocity, then velocity is the slope of the position vs time graph (rise over run)
• If the slope is positive, the object is moving in the +x direction.
• If the slope is negative, the object is moving in the −x direction.
• The magnitude of the slope represents the speed of the object.

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• The speed and the direction together are called the velocity of the object.

Speed of “The Wave”

00

~70 seats around the circle

Speed of “The Wave”

• Δx:
• Δ t:
• v_x:

One of the most important equations in this course: constant velocity:

Let’s assume the wave does not accelerate.

(This is a simplification.)

Section 2.3. Position-versus-Time Graphs

Position [m]

1

Time [seconds]

2

3

4

5

6

7

8

9

10

7

Home

• Imagine you steadily walked 7 m away from your house in 5 seconds, then you stopped for 2 seconds to check your bag, then you ran back to your house in 3 seconds.

Section 2.3. Position-versus-Time Graphs

Time [seconds]

Position [cm]

5

10

20

10

Steeper Slope ⇒ Faster Motion

Section 2.3. Position-versus-Time Graphs

Shallower Slope

⇒ Slower Motion

Time [seconds]

Position [cm]

5

10

20

10

Section 2.3. Position-versus-Time Graphs

Zero Slope

⇒ Object is not moving

Time [seconds]

Position [cm]

20

10

5

10

You hike two thirds of the way to the top of a hill at a speed of v₁, and run the final third at a speed of v₂. What was your average speed?

SKETCH & TRANSLATE.

SIMPLIFY & DIAGRAM

REPRESENT MATHEMATICALLY

SOLVE & EVALUATE

Graphical method for finding displacement

• The slope of a position vs time graph gives the velocity at any time. [This is called a “derivative” in calculus, but you don’t need to know that for this course.]
• The area under a velocity vs time graph gives the displacement over a certain time interval. [This is called an “integral” in calculus, but you don’t need to know that for this course.]

Curved Line = Not-Constant Velocity

x

t

v

t

Last day I asked at the end of class:

1. Does constant velocity imply constant acceleration?

Yes! And, in fact, it implies the acceleration is zero (which is a constant).

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2. Does constant acceleration imply constant velocity?

No! (unless this constant happens to be zero)… Constant acceleration implies the velocity is changing at a constant rate.

“Deceleration”

• Can we quickly go over the difference between negative acceleration and deceleration?
• Can you briefly clarify how deceleration and negative acceleration can sometimes be the same and sometimes not?
• Can you clarify deceleration and negative acceleration? I am confused with these two.

• I avoid the word “deceleration”. Instead, I just say “acceleration”, which, in physics, is more general, and includes deceleration as a special case.
• Acceleration is the important concept. If an object’s velocity is changing, I simply say it is accelerating. It might be speeding up, slowing down, turning, etc. In all cases it has acceleration, and the SI unit of acceleration is m/s².
• If an object is “decelerating”, it just means that it is slowing down.

The sign of acceleration

• When you are doing a problem, the first step is to DRAW A DIAGRAM.
• On this diagram, you can draw your coordinate axes, showing what you define as the +x direction.
• If something is moving in your +x direction, its velocity is positive.
• If something is moving opposite to your +x direction, its velocity is negative.
• If something is moving in a straight line and speeding up, its acceleration has the same sign as the velocity.
• If something is moving in a straight line and slowing down, its acceleration has the opposite sign as the velocity.

Demonstration

From the preclass survey

• “So is there a rate of change of acceleration?”
• Yes! You can take the slope of an acceleration vs time graph and get a value called “jerk” in m/s³.
• But jerk is not studied in this course because it is not very interesting and doesn’t relate to anything.
• Acceleration is extremely important because it is directly proportional to the net-force on an object, and inversely proportional to an object’s mass (as we’ll learn in Chapter 4)

The 3 Most Commonly used Equations of Constant Acceleration:

Strategy: When a = constant, you can use one of these equations. Figure out which variable you don’t know and don’t care about, and use the equation which doesn’t contain it.

v

t

10 m/s

5 s

• For 1D motion with constant acceleration, the average velocity is ½(initial velocity + final velocity)

Average Velocity

The 4 Equations of Constant Acceleration:

Strategy: When a = constant, you can use one of these equations. Figure out which variable you don’t know and don’t care about, and use the equation which doesn’t contain it.

Newton’s First Law of motion

For an observer in an inertial reference frame, when the net force on an object is zero, the object stays at rest or continues moving at a constant velocity.

1

Relative Motion

If we know Object A’s velocity measured in one reference frame, 1, we can transform it into the velocity that

would be measured by an experimenter in a different reference frame, 2, using the Galilean transformation of velocity.

Joke: Why Did the Chicken Cross the Road?

Aristotle (330 BC):

“Because it is the nature of chickens to cross roads.”

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Newton (1687):

“Because there is no external net force causing the chicken’s velocity across the road to change.”

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Einstein (1905):

“Is the chicken crossing the road, or is the road moving under the chicken?”

[image downloaded 9/30/2013 from http://afgg.tumblr.com/post/3126636261/when-the-chicken-crossed-the-road ]

The 4 Equations of Constant Acceleration:

Strategy: When a = constant, you can use one of these equations. Figure out which variable you don’t know and don’t care about, and use the equation which doesn’t contain it.

Example 1. An Olympic-class sprinter starts a race with an acceleration of 4.50 m/s². What is her speed 2.40 s later?

Example 2. At the end of a race, a runner decelerates from a velocity of 9.00 m/s at a rate of 2.00 m/s². How far does she travel in the next 4.00 s?

Example 3. You are traveling at an initial velocity of v₀ = 30 m/s. At position x₀ = 0 you apply the brakes, which gives you a constant acceleration of a = –10 m/s². How far do you travel before stopping?

Example 4. A motorcycle accelerates from rest to 26.8 m/s in 3.90 s. Assuming constant acceleration, how far does it travel in that time?.

“Thought Experiment”

1 brick

Mass = 2 kg

• If you drop one brick with a mass of 2 kg, it will fall at a certain rate.
• If you drop a double-sized brick with a mass of 4 kg, should it fall faster?

• If so, what if you drop two bricks that are connected by a tiny, very strong wire, so they actually form one 4 kg object?

Free Fall

= Falling under the influence of gravity only, with no air resistance.

• Freely falling objects on Earth accelerate at the rate of 9.8 m/s/s, i.e., 9.8 m/s²
• The exact value of free fall acceleration depends on altitude and latitude on the earth.
• For this course, let’s use g = 9.80 m/s²

Free Fall—How Fast?

The velocity acquired by an object starting from rest is

So, under free fall, when acceleration is 9.8 m/s², the speed is

• 9.8 m/s after 1 s.
• 19.6 m/s after 2 s.
• 29.4 m/s after 3 s.

And so on.

• Even though the force of gravity of the Earth on the basketball is more than on the tennis ball, the balls accelerate downward at the same rate.
• This was Galileo’s amazing discovery.
• In 1633 the Roman Inquisition found him “gravely suspect of heresy” for his scientific findings, and sentenced him to house arrest, under which he remained until his death, 9 years later.

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Free Fall

This morning’s preclass survey

From this morning’s preclass survey

• Question: After the object hits the ground, will it bounce back?
• Harlow Answer: Maybe! Actual motion is complicated. If a ball is thrown there is a force from the hand, and there is the force from the ground when it hits. These equations of constant acceleration only apply to AFTER the ball leaves the hand and BEFORE the ball touches the ground.
• Question: Are these equations basically kinematic equation in vertical direction?
• Harlow answer: Yes! Just replace x with y, and a with –g!
• Question: Can human live in a planet that doesn’t have a as 9.81?
• Harlow answer: Yes, probably! Mars has 3.17, so let’s hope so!

Air resistance

• Why does a crumpled up piece of paper fall faster than a non-crumpled paper? Which is more massive?
• The non-crumpled paper has a large surface area which catches a lot of air, like a sail, as it falls.
• This produces an upward force of air resistance on the non-crumpled paper that cannot be neglected.
• So the non-crumpled paper is not in freefall, and it does not fall with acceleration 9.8 m/s². It falls much more slowly.
• This is true of many light objects such as feathers, balloons, tissues, etc.

Example.

Some people in a hotel are dropping water balloons from their open window onto the ground below. The balloons take 0.15 s to pass your 1.6-m-tall window. Where should security look for the raucous hotel guests?

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2.54

2.53

Graphical method for finding displacement

• The slope of a position vs time graph gives the velocity at any time. [This is called a “derivative” in calculus, but you don’t need to know that for this course.]
• The area under a velocity vs time graph gives the displacement over a certain time interval. [This is called an “integral” in calculus, but you don’t need to know that for this course.]

Here is the velocity graph of an object that is at the origin (x = 0 m) at t = 0 s. At t = 4.0 s, the object’s position is?

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