motion. chapter four: motion 4.1 position, speed and velocity 4.2 graphs of motion 4.3...

MOTION

Chapter Four: Motion

4.1 Position, Speed and Velocity

4.2 Graphs of Motion4.3 Acceleration

4.1 Position, Speed and VelocityPosition is a variable given relative

to an origin.The origin is the place where position equals 0.

The position of this car at 50 cm describes where the car is relative to the track.

4.1 Position, Speed and VelocityPosition and distance are similar

but not the same.If the car moves a distance of 20 cm to the right, its new position will be 70 cm from its origin.

Distance = 20 cm

New position

4.1 Position, Speed and VelocityThe variable speed describes how

quickly something moves. To calculate the speed of a moving object divide the distance it moves by the time it takes to move.

4.1 Position, Speed and VelocityThe units for speed are distance

units over time units.This table shows different units commonly used for speed.

4.1 Average speed

When you divide the total distance of a trip by the time taken you get the average speed.

On this driving trip around Chicago, the car traveled and average of 100 km/h.

4.1 Instantaneous speed

A speedometer shows a car’s instantaneous speed.

The instantaneous speed is the actual speed an object has at any moment.

How far do you go if you drive for two hours at a speed of 100 km/h?

1. Looking for: …distance

2. Given: …speed = 100 km/h time = 2 h

3. Relationships: d = vt

4. Solution: d = 100 km/h x 2 h = 200 km

= 200 km

Solving Problems

4.1 Vectors and velocity Position uses positive and negative

numbers. Positive numbers are for positions to

the right of the origin and negative numbers are for positions to the left the origin.

4.1 Vectors and velocity

Distance is either zero or a positive value.

4.1 Vectors and velocity We use the term velocity to

mean speed with direction.

4.1 Keeping track of where you are Pathfinder is a small robot sent

to explore Mars.

It landed on Mars in 1997.

Where is Pathfinder now?

4.1 Keeping track of where you are Pathfinder keeps track of its

velocity vector and uses a clock. Suppose Pathfinder moves

forward at 0.2 m/s for 10 seconds.

What is Pathfinder’s velocity?

4.1 Keeping track of where you are Suppose Pathfinder goes

backward at 0.2 m/s for 4 seconds.

What is Pathfinder’s change in position?

4.1 Keeping track of where you areThe change in position is the

velocity multiplied by the time.

4.1 Keeping track of where you areEach change in position is added

up using positive and negative numbers.

Pathfinder has a computer to do this.

4.1 Maps and coordinates If Pathfinder was crawling on a

straight board, it would have only two choices for direction.

Out on the surface of Mars, Pathfinder has more choices.

The possible directions include north, east, south, and west, and anything in between.

4.1 Maps and coordinates A graph using north−south and

east−west axes can accurately show where Pathfinder is.

This kind of graph is called a map.

Street maps often use letters and numbers for coordinates.

4.1 Vectors on a map Suppose you run east for 10

seconds at a speed of 2 m/s. Then you turn and run south at the

same speed for 10 more seconds.

Where are you compared to where you started?

4.1 Vectors on a mapTo get the answer, you figure out your east−west changes and your north−south changes separately.origin = (0 , 0)

4.1 Vectors on a mapYour first

movement has a velocity vector of +2 m/s, west-east (x-axis).

After 10 seconds your change in position is +20 meters (east on x-axis).

d = v x t d = 2 m/s x 10 s = +20 m

4.1 Vectors on a mapYour second

movement has a velocity vector of −2 m/s north−south (y-axis)

In 10 seconds you move −20 meters (south is negative on y-axis)

d = 2 m/s x 10 s = -20 m New position = (+20 , -20)

A train travels at 100 km/h heading east to reach a town in 4 hours. The train then reverses and heads west at 50 km/h for 4 hours. What is the train’s position now?

1. Looking for: …train’s new position

2. Given: …velocity = +100 km/h, east ; time = 4 h …velocity = -50 km/h, west ; time = 4 h

3. Relationships: change in position = velocity × time

Solving Problems

4. Solution: 1st change in position:

(+100 km/h) × (4 h) = +400 km

2nd change in position: (−50 km/h) × (4 h) = −200 km

Final position: (+400 km) + (−200 km) = +200 km The train is 200 km east of where it started.

Solving Problems

2.3 Graphing A graph is a visual way to

organize data.

A scatterplot or XY graph is used to see if two variables are related.

2.3 Graphing

A bar graph compares data grouped by a name or category.

2.3 Graphing A pie graph

shows the amount each part makes of up of the whole (100%).

2.3 Graphing A “connect-the-dots” line graph

is often used to show trends in data over time.

2.3 How to make an XY graphScatterplots show how a change in one

variable influences another variable.The independent variable is the

variable you believe might influence another variable.

The dependent variable is the variable that you hope will change as a result of the experiment.

2.3 How to make an XY graphPressure is critical to safe diving.How does an increase in depth affect the pressure?

What sort of graph would best show the relationship between pressure and depth?

2.3 How to make an XY graph

1. Choose x and y-axis Depth is the independent variable = x axis Pressure is the dependent variable = y

axis2. Make a scale Most graphs use ones, twos, fives or tens OR calculate the value per box

3. Plot your data 4. Create a title* Exception- when time is a variable

2.3 Identifying graph relationshipsIn a direct

relationship, when one variable increases, so does the other.The speed and distance variables show a direct relationship.

2.3 Identifying graph relationships

When there is no relationship the graph looks like a collection of dots.

No pattern appears.

2.3 Identifying graph relationshipsIn an

inverse relationship, when one variable increases, the other decreases.

2.3 Reading a graphWhat is the speed of the car at 50 cm?1. Find the known value on the x axis

Position = 50 cm

2. Draw a line vertically upward from 50 cm until it hits the curve.

3. Draw a line across horizontally to the y-axis from the same place on the curve.

4. Read the speed using the y axis scale. Speed = 76 cm/s

4.2 Graphs of Motion Constant speed means the speed

stays the same. An object moving at a constant speed

always creates a position vs. time graph that is a straight line.

4.2 Graphs of Motion The data shows

the runner took 10 seconds to run each 50-meter segment.

Because the time was the same for each segment, you know the speed was the same for each segment.

4.2 Graphs of Motion You can use

position vs. time graphs to compare the motion of different objects.

The steeper line on a position vs. time graph means a faster speed.

4.2 Slope The steepness of a line is

measured by finding its slope. The slope of a line is the ratio of

the “rise” to the “run”.

4.2 Graphs of changing motion Objects rarely move

at the same speed for a long period of time.

A speed vs. time graph is also useful for showing the motion of an object that is speeding up or slowing down.

4.2 Graphs of changing motion Suppose we draw a rectangle on

the speed vs. time graph between the x-axis and the line showing the speed.

The area of the rectangle is equal to its length times its height.

On the graph, the length is equal to the time and the height is equal to the speed.

4.3 Curved motion

Circular motion is another type of curved motion.

An object in circular motion has a velocity vector that constantly changes direction.

4.3 AccelerationAcceleration is the rate at which your speed (or velocity) changes.

4.3 AccelerationWhat is the bike’s acceleration?

4.3 AccelerationAcceleration describes how quickly speed changes.

Acceleration is the change in speed divided by the change in time.

4.3 Speed and accelerationAn acceleration of 20 km/h/s means that the speed increases by 20 km/h each second.

The units for time in acceleration are often expressed as “seconds squared” and written as s2.

Can you convert this rate using conversion

factors?

Solving Problems

A sailboat moves at 1 m/s.

A strong wind increases its speed to 4 m/s in 3 s.

Calculate acceleration.

1. Looking for: …acceleration of sailboat

2. Given: …v1 = 1 m/s; v2 = 4 m/s; time = 3 s

3. Relationships: a = v2 – v1/t

4. Solution: a = (4 m/s – 1 m/s)/ 3 s

= 1 m/s2

Solving Problems

4.3 Acceleration on speed-time graphs

Positive acceleration adds more speed each second.

Things get faster.

Speed increases over time.

4.3 Acceleration on speed-time graphs

Negative acceleration subtracts some speed each second.

Things get slower. People sometimes use the word deceleration to describe slowing down.

4.3 Acceleration on position-time graphs

The position vs. time graph is a curve when there is acceleration.

The car covers more distance each second, so the position vs. time graph gets steeper each second.

4.3 Acceleration on position-time graphs

When a car is slowing down, the speed decreases so the car covers less distance each second.

The position vs. time graph gets shallower with time.

4.3 Free fallAn object is in free fall if it is accelerating due to the force of gravity and no other forces are acting on it.

4.3 Free fallFalling objects increase their speed by 9.8 m/s every second, or 9.8 m/s2

4.3 Acceleration and direction A car driving around a curve at

a constant speed is accelerating because its direction is changing.

4.3 Curved motionA soccer ball is an

example of a projectile.

A projectile is an object moving under the influence of only gravity.

The path of the ball makes a bowl-shaped curve called a parabola.

4.3 Curved motion

Circular motion is another type of curved motion.

An object in circular motion has a velocity vector that constantly changes direction.