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FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/EPHYSICS

RANDALL D. KNIGHT

Chapter 1 Lecture

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Four Basic Types of Motion

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An easy way to study motion is to make a videoof a moving object.

A video camera takes images at a fixed rate, typically 30 every second.

Each separate image is called a frame.

Shown are four frames from a video of a car going past.

The car is in a somewhat different position in each frame.

Making a Motion Diagram

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Suppose we edit the video by layering the frames on top of each other, creating the composite image shown below.

This edited image, showing an object’s position at several equally spaced instants of time, is called a motion diagram.

Making a Motion Diagram

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Images that are equally spaced indicate an object moving with constant speed.

Examples of Motion Diagrams

An increasing distance between the images shows that the object is speeding up.

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A decreasing distance between the images shows that the object is slowing down.

Examples of Motion Diagrams

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For many types of motion, we can treat the object as if all its mass were concentrated into this single point.

An object that can be represented as a mass at a single point in space is called a particle.

If we model an object as a particle, we can represent the object in each frame of a motion diagram as a simple dot rather than having to draw a full picture.

Below is such a motion diagram of a car stopping.

The Particle Model

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Motion diagram of a rocket launch

The Particle Model

Motion Diagram in which the object is represented as a particle

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To use a motion diagram, you would like to know where the object is and when the object was at that position.

Position measurements can be made by laying a coordinate-system grid over a motion diagram.

To illustrate, the figure shows a sled sliding down a snow-covered hill.

(b) shows a motion diagram for the sled, over which we’ve drawn an xy-coordinate system.

Position and TimeVectors and Scalars

Vector a quantity that has a direction as well as a magnitude

Scalar a quantity that only has a magnitude (number and units)

ex: position, displacement, velocity, acceleration, force, momentum

ex: distance, speed, time, mass, energy, work, density, temperature

Vectors

vectors are represented with arrows

the length of the vector is proportional to the magnitude of the vector (use an appropriate scale such as 1 cm = 10 N)

the direction of the arrow is the direction of the vector

100 N force at 45o 50 N force at 0o

tail

tip

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Tactics: Vector Addition

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Adding Vectors Graphically

1) Draw one of the vectors to scale and in the right direction.

Tail-to-tip method:

choose an appropriate scale (1 in. = 25 N)

2) Draw the second vector (to scale and in the right direction) starting from the tip of the first.

3) The resultant vector is drawn from the tail of the first vector to the tip of the second vector.

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Tactics: Vector Subtraction

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We said that motion is the change in an object’s position with time, but how do we show a change of position?

Shown is the motion diagram of a sled sliding down a snow-covered hill.

Displacement

To show how the sled’s position changes between t3 = 3 s and t4 = 4 s, we draw a vector arrow between the two dots of the motion diagram.

This vector is the sled’s displacement, which is given the symbol Δr.

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Different observers may choose different coordinate systems and different clocks, however, all observers find the same values for the displacement Δ and the time interval Δt.

Time Interval

A stopwatch is used to measure a timeinterval.

It’s useful to consider a change in time.

An object may move from an initial position at time ti to a final position at time tf.

The time interval is Δt = tf − ti.

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To quantify an object’s fastness or slowness, we define a ratio:

Average speed does not include information about direction of motion.

The average velocity of an object during a time interval Δt, in which the object undergoes a displacement Δ , is the vector:

Average Speed, Average Velocity

The victory goes to the runner with the highest average speed.

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The velocity vector is in the same direction as the displacement Δ

The length of is directly proportional to the length of Δ .

Consequently, we may label the vectors connecting the dots on a motion diagram as velocity vectors .

Below is a motion diagram for a tortoise racing a hare.

Motion Diagrams with Velocity Vectors

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EXAMPLE 1.2 Accelerating Up a Hill

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Sometimes an object’s velocity is constant as it moves.

More often, an object’s velocity changes as it moves.

Acceleration describes a change in velocity.

Consider an object whose velocity changes from to during the time interval ∆t.

The quantity is the change in velocity.

The rate of change of velocity is called the average acceleration:

Linear Acceleration

The Audi TT accelerates from 0 to 60 mph in 6 s.

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Tactics: Finding the Acceleration Vector

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Notice that the acceleration vector goes beside the dots, not beside the velocity vectors.

That is because each acceleration vector is the difference between two velocity vectors on either side of a dot.

Tactics: Finding the Acceleration Vector

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The Complete Motion Diagram

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Example 1.5 Skiing Through the Woods

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Example 1.5 Skiing Through the Woods

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When an object is speeding up, the acceleration and velocity vectors point in the same direction.

When an object is slowing down, the acceleration and velocity vectors point in opposite directions.

An object’s velocity is constant if and only if its acceleration is zero.

In the motion diagrams to the right, one object is speeding up and the other is slowing down, but they both have acceleration vectors toward the right.

Speeding Up or Slowing Down?

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Tactics: Determining the Sign of the Position, Velocity, and Acceleration

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Tactics: Determining the Sign of the Position, Velocity, and Acceleration

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Tactics: Determining the Sign of the Position, Velocity, and Acceleration

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Below is a motion diagram, made at 1 frame per minute, of a student walking to school.

A motion diagram is one way to represent the student’s motion.

Another way is to make a graph of x versus t for the student:

Position-versus-Time Graphs

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Example 1.7 Interpreting a Position Graph

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Example 1.7 Interpreting a Position Graph

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A new building requires careful planning. The architect’s visualization and drawings have to be complete before the detailed procedures of construction get under way. The same is true for solving problems in physics.

Physics problems are often presented using words, which can be imprecise or ambiguous.

Part of problem-solving involves using symbols and drawings to create a representation, which is clear and precise.

A verbal representation is a problem statement or re-statement using words.

A pictorial representation includes motion diagrams, coordinate systems, simple drawings, and symbols.

A graphical representation uses graphs when appropriate.

A mathematical representation uses specific equations which must be solved.

Solving Problems in Physics

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Tactics: Drawing a Pictorial Representation

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Tactics: Drawing a Pictorial Representation

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General Problem-Solving Strategy

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Example 1.9 Launching a Weather Rocket

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Example 1.9 Launching a Weather Rocket

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Example 1.9 Launching a Weather Rocket

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Example 1.9 Launching a Weather Rocket

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Science is based on experimental measurements, and measurements require units.

The system of units in science is called le SystèmeInternationale d’unités or SI units.

The SI unit of time is the second, abbreviated s.

1 s is defined as the time required for 9,192,631,770 oscillations of the radio wave absorbed by a cesium-133 atom.

The SI unit of length is the meter, abbreviated m.

1 m is defined as the distance traveled by light in a vacuum during 1/299,292,458 of a second.

Units

An atomic clock at the National Institute of Standards and Technology is the primary standard of time.

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The SI unit of mass is the kilogram, abbreviated kg.

1 kg is defined as the mass of the international standard kilogram, a polished platinum-iridium cylinder stored in Paris.

Many lengths, times, and masses are either much less or much greater than the standards of 1 m, 1 s, and 1 kg.

We use prefixes to denote various powers of 10, which make it easier to talk about quantities.

Units

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It is important to be able to convert back and forth between SI units and other units.

One effective method is to write the conversion factor as a ratio equal to one.

Because multiplying by 1 does not change a value, these ratios are easily used for unit conversions.

For example, to convert the length 2.00 feet to meters, use the ratio

So that

Unit Conversions

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When problem solving, it is important to decide whether or not your final answer “makes sense.”

For example, if you are working a problem about automobile speeds and reach an answer of 35 m/s, is this a realistic speed?

The table shows some approximate conversion factors that can be used to assess answers.

Using 1 m/s ≈ 2 mph, you find that 35 m/s is roughly 70 mph, a reasonable speed for a car.

If you reached an answer of 350 m/s, this would correspond to an unreasonable 700 mph, indicating that perhaps you made a calculation error.

Assessment

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Chapter 1 Summary Slides

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General Strategy

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General Strategy

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Important Concepts

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Important Concepts