4.1 Position, Speed, and Velocityskovranscience.weebly.com/uploads/8/5/1/8/8518324/___chpt_4.pdf ·...

78 UNIT 2 MOTION AND FORCE

Chapter 4 MOTION

Figure 4.1: If the car moves 20 cm to the right, its position will be 70 cm.

position - a variable that tells location relative to an origin.

origin - a place where the position has been given a value of zero.

4.1 Position, Speed, and VelocityWhere are you right now? How fast are you moving? To answer these questions precisely, you need to use the concepts of position, speed, and velocity. These ideas apply to ordinary objects, such as cars, bicycles, and people. They also apply to microscopic objects the size of atoms and to enormous objects like planets and stars. Let’s begin our discussion of motion with the concept of position.

The position variablePosition as a

variableYou may do an experiment in your class that uses a car on a track. How do you tell someone exactly where the car is at any given moment? The answer is by measuring its position. Position is a variable. The position of the car describes where the car is relative to the track. In the diagram below, the position of the car is 50 centimeters (cm). That means the center of the car is at the 50 cm mark on the track.

Position anddistance

Position and distance are similar but not the same. Both use units of length. However, position is given relative to an origin. The origin is the place where position equals 0 (near the left end of the track above). Here’s an example of the difference between position and distance. Assume the track is 1 meter long. Suppose the car moves a distance of 20 cm away from the 50 cm mark. Where is it now? You know a distance (20 cm) but you still don’t know where the car is. It could have moved 20 cm to the right or 20 cm to the left. Saying the car is at a position of 70 cm tells you where the car is. A position is a unique location relative to an origin (Figure 4.1).

794.1 POSITION, SPEED, AND VELOCITY

Chapter 4MOTION

SpeedSpeed is a motion

variableThe variable speed describes how quickly something moves. To calculate the speed of a moving object, you divide the distance it moves by the time it takes to move. For example, if you drive 120 miles (the distance) and it takes you 2 hours (the time) your speed is 60 miles per hour (60 mph = 120 miles ÷ 2 hours). The lower case letter v is used to represent speed.

Units for speed The units for speed are distance units over time units. If distance is in kilometers and time in hours, then speed is in kilometers per hour (km/h). Other metric units for speed are cm per second (cm/s) and meters per second (m/s). Your family’s car probably shows speed in miles per hour (mph). Table 4.1 shows different units commonly used for speed.

Average speed andconstant speed

When you divide the total distance of a trip by the time taken, you get the average speed. Figure 4.2 shows an average speed of 100 km/h. But, think about actually driving though Chicago. On a real trip, your car will slow down and speed up. Sometimes your speed will be higher than 100 km/h, and sometimes lower (even 0 km/h!) The speedometer shows you the car’s instantaneous speed. The instantaneous speed is the actual speed an object has at any moment.

Table 4.1: Common Units for Speed

Distance Time Speed Abbreviationmeters seconds meters per second m/s

kilometers hours kilometers per hour km/hcentimeters seconds centimeters per second cm/s

miles hours miles per hour mph

Figure 4.2: A driving trip with an average speed of 100 km/h.

speed - describes how quickly an object moves, calculated by dividing the distance traveled by the time it takes.

average speed - the total distance divided by the total time for a trip.

instantaneous speed - the actual speed of a moving object at any moment.


Chapter 4 MOTION

The Speed Limit of the Universe

The fastest speed in the universe is the speed of light. Light moves at 300 million meters per second (3 x 108 m/s). If you could make light travel in a circle, it would go around the Earth 7.5 times in one second! Scientists believe the speed of light is the ultimate speed limit in the universe.

a. Your grandmother’s house is 60 km away from where you started.

b. The snake’s speed is 4 m/s.

c. It will take the train 12 hours to travel 600 kilometers.

Solving Problems: Speed

How far will you go if you drive for 2 hours at a speed of 100 km/h?

1. Looking for: You are asked for a distance.

2. Given: You are given the speed and the time.

3. Relationships: distance = speed × time

4. Solution: distance = (100 km/h) × (2 h) = 200 km

Your turn...

a. You travel at an average speed of 20 km/h in a straight line to get to your grandmother’s house. It takes you 3 hours to get to her house. How far away is her house from where you started?

b. What is the speed of a snake that moves 20 meters in 5 seconds?

c. A train is moving at a speed of 50 km/h. How many hours will it take the train to travel 600 kilometers?


Chapter 4MOTION

Vectors and velocityTelling “in front”

from “behind”How can you tell the difference between one meter in front of you and one meter behind you? The variable of distance is not the answer. The distance between two points can only be positive (or zero). You can’t have a negative distance. For example, the distances between the ants in Figure 4.3 are either positive or zero. Likewise, one meter in front of you and one meter behind you both have the same distance: 1 meter.

Using positive andnegative numbers

The answer is to use position and allow positive and negative numbers. In the diagram below, positive numbers describe positions to the right (in front) of the origin. Negative numbers are to the left (or behind) the origin.

Vectors Position is an example of a kind of variable called a vector. A vector is a variable that tells you a direction as well as an amount. Positive and negative numbers are enough information for a variable when the only directions are forward and backward. When up−down and right−left are also possible directions, vectors get more complicated.

Velocity Like position, motion can go right, left, forward or backward. We use the term velocity to mean speed with direction. Velocity is positive when moving to the right, or forward. Velocity is negative when moving to the left, or backward (Figure 4.4).

The differencebetween velocity

and speed

Velocity is a vector, speed is not. In regular conversation you might use the two words to mean the same thing. In science, they are related but different. Speed can have only a positive value (or zero) that tells you how far you move per unit of time (like meters per second). Velocity is speed and direction. If the motion is in a straight line, the direction can be shown with a positive or negative sign. The sign tells whether you are going forward or backward and the quantity (speed) tells you how quickly you are moving.

Figure 4.3: Distance is always a positive value or zero.

Figure 4.4: Velocity can be a positive or a negative value.

vector - a variable that gives direction information included in its value.

velocity - a variable that tells you both speed and direction.


Chapter 4 MOTION

Figure 4.5: Pathfinder is a robot explorer which landed on Mars in 1997 (NASA/JPL).

Figure 4.6: The change in position or distance is the velocity multiplied by the time.

Figure 4.7: Each change in position is added up using positive and negative numbers.

Keeping track of where you areA robot uses vectors Pathfinder is a small robot sent to explore Mars (Figure 4.5). As it moved,

Pathfinder needed to keep track of its position. How did Pathfinder know where it was? It used its velocity vector and a clock to calculate every move it made.

Use two variables tofind the third one

Any formula that involves speed can also be used for velocity. For example, you move 2 meters if your speed is 0.2 m/s and you keep going for 10 seconds. But did you move forward or backward? You move −2 meters (backwards) if you move with a velocity of −0.2 m/s for 10 seconds. Using the formulas with velocity gives you position instead of distance.

Forward andbackward movement

Suppose Pathfinder moves forward at 0.2 m/s for 10 seconds. Its velocity is +0.2 m/s. In 10 seconds, its position changes by +2 meters.

Now, suppose Pathfinder goes backward at 0.2 m/s for 4 seconds. This time the velocity is −0.2 m/s. The change in position is −0.8 meters. A change in position is velocity × time (Figure 4.6).

Adding up a seriesof movements

The computer in Pathfinder adds up +2 m and −0.8 m to get +1.2 m. After these two moves, Pathfinder’s position is 1.2 meters in front of where it was. Pathfinder knows where it is by keeping track of each move it makes. It adds up each change in position using positive and negative numbers to come up with a final position (Figure 4.7).


Chapter 4MOTION

Maps and coordinatesTwo dimensions If Pathfinder was crawling on a straight board, it would have only two choices

for direction. Positive is forward and negative is backward. Out on the surface of Mars, Pathfinder has more choices. It can turn and go sideways! The possible directions include north, east, south, west, and anything in between. A flat surface is an example of two dimensions. We say two because it takes two number lines to describe every point (Figure 4.8).

North, south, east,and west

One way to describe two dimensions is to use north−south as one number line, or axis. Positive positions are north of the origin. Negative positions are south of the origin. The other axis goes east−west. Positive positions on this axis are east of the origin. Negative positions are west of the origin.

Coordinatesdescribe position

Pathfinder’s exact position can be described with two numbers. These numbers are called coordinates. The graph at the left shows Pathfinder at the coordinates of (4, 2) m. The first number (or coordinate) gives the position on the east−axis. Pathfinder is 4 m east of the origin. The second number gives the position on the north−south axis. Pathfinder is 2 m north of the origin.

Maps A graph using north−south and east−west axes can accurately show where Pathfinder is. The graph can also show any path Pathfinder takes, curved or straight. This kind of graph is called a map. Many street maps use letters on the north−south axis and numbers for the east−west axis. For example, the coordinates F-4 identify the square that is in row F, column 4 of the map shown in Figure 4.9.

Figure 4.8: A flat surface has two perpendicular dimensions: north−south and east−west. Each dimension has positive and negative directions.

Figure 4.9: Street maps often use letters and numbers for coordinates.

axis - one of two (or more) number lines that form a graph.

coordinates - values that give a position relative to an origin.


Chapter 4 MOTION

Figure 4.10: A running trip with a turn.

Captain Vector’s Hidden TreasureUse these velocity vectors to determine the location of Captain Vector’s hidden pirate treasure. Your starting place is (0, 0).

1. Walk at a velocity of 1 m/s south for 10 seconds.

2. Then, jog at a velocity of 3 m/s east for 5 seconds.

3. Run at a velocity of 5 m/s north for 2 seconds.

4. Then walk backward south at a velocity of 0.5 m/s for 2 seconds.

Where is the treasure relative to your starting place?

Vectors on a mapA trip with a turn 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 (Figure 4.10). Where are you compared to where you started? To get the answer, you figure out your east−west changes and your north−south changes separately.

Figure eachdirection separately

Your first movement has a velocity vector of +2 m/s on the east−west axis. After 10 seconds your change in position is +20 meters (east). There are no more east−west changes because your second movement is north−south only. Your second movement has a velocity vector of −2 m/s north−south. In 10 seconds you moved −20 meters. The negative sign means you moved south.

Figuring your finalposition

Now add up any east−west changes to get your final east−west position.Do the same for your north−south position. Your new position is (+20 m, −20 m).


Chapter 4MOTION

Solving Problems: Velocity

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: You are asked for position.

2. Given: You are given two velocity vectors and the times for each.

3. Relationships: change in position = velocity × time

4. Solution: The first change in position is (+100 km/h) × (4 h) = +400 kmThe second change in position is (−50 km/h) × (4 h) = −200 kmThe final position is (+400 km) + (−200 km) = +200 km. The train is 200 km east of where it started.

Your turn...a. You are walking around your town. First you walk north from your starting

position and walk for 2 hours at 1 km/h. Then, you walk west for 1 hour at 1 km/h. Finally, you walk south for 1 hour at 2 km/h. What is your new position relative to your starting place?

b. A ship needs to sail to an island that is 1,000 km south of where the ship starts. If the captain sails south at a steady velocity of 30 km/h for 30 hours, will the ship make it?

Fast Trains!The Bullet train of Japan was the world’s first high-speed train. When it came into use in 1964, it went 210 km/h.

Research today’s high-speed trains of the world. How fast they go?

Research to find out why the United States lags behind in having high-speed trains. Find out the advantages and disadvantages of having high-speed trains in the U.S..

a. Your new position is 1 kilometer west of where you started.

b. No, because 30 km/h × 30 h = 900 km. The island is still 100 km away.


Chapter 4 MOTION

Look at the graphic below and answer the following questions.

1. How fast is each cyclist going in units of meters per second*?

2. Which cyclist is going faster? How much faster is this cyclist going compared to the other one?

*The word per means “for every” or “for each.” Saying “5 kilometers per hour” is the same as saying “5 kilometers for each hour.” You can also think of per as meaning “divided by.” The quantity before the word per is divided by the quantity after it.

Section 4.1 Review1. What is the difference between distance and position? 2. From an origin you walk 3 meters east, 7 meters west, and then 6 meters

east. Where are you now relative to the origin?3. What is your average speed if you walk 2 kilometers in 20 minutes?4. Give an example where instantaneous speed is different from average

speed.5. A weather report says winds blow at 5 km/h from the northeast. Is this

description of the wind a speed or velocity? Explain your answer.6. What velocity vector will move you 200 miles east in 4 hours traveling

at a constant speed?7. Give an example of a situation in which you would describe an object’s

position in: a. one dimensionb. two dimensionsc. three dimensions

8. A movie theater is 4 kilometers east and 2 kilometers south of your house.a. Give the coordinates of the movie theater. Your house is the origin.b. After leaving the movie theater, you drive 5 kilometers west and

3 kilometers north to a restaurant. What are the coordinates of the restaurant? Use your house as the origin.

874.2 GRAPHS OF MOTION

Chapter 4MOTION

4.2 Graphs of MotionConsider the phrase “a picture is worth a thousand words.” A graph is a special kind of picture that can quickly give meaning to a lot of data (numbers). You can easily spot relationships on a graph. It is much more difficult to see these same relationships in columns of numbers. Compare the table of numbers to the graph in Figure 4.11 and see if you agree!

The position vs. time graphRecording data Imagine you are helping a runner who is training for a track meet. She wants

to know if she is running at a constant speed. Constant speed means the speed stays the same. You mark the track every 50 meters. Then you measure her time at each mark as she runs. The data for your experiment is shown in Figure 4.11. This is position vs. time data because it tells you the runner’s position at different points in time. She is at 50 meters after 10 seconds,100 meters after 20 seconds, and so on.

Graphing data To graph the data, you put position on the vertical (y) axis and time on the horizontal (x) axis. Each row of the data table makes one point on the graph. Notice the graph goes over to the right 10 seconds and up 50 meters between each point. This makes the points fall exactly in a straight line. The straight line tells you the runner moves the same distance during each equal time period. An object moving at a constant speed always creates a straight line on a position vs. time graph.

Calculating speed The data shows that the runner took 10 seconds to run each 50-meter segment. Because the time and distance was the same for each segment, you know her speed was the same for each segment. You can use the formula v = d/t to calculate the speed. Dividing 50 meters by 10 seconds tells you her constant speed was 5 meters per second.

Figure 4.11: A data table and a position vs. time graph for a runner.

constant speed - speed that stays the same and does not change.


Chapter 4 MOTION

Figure 4.12: This graph shows that the average speed between A and B increases as the angle of the track goes from 10 degrees to 50 degrees.

Graphs show relationships between variablesRelationships

between variablesThink about rolling a toy car down a ramp. You theorize that steeper angles on the ramp will make the car go faster. How do you find out if your theory is correct? You need to know the relationship between the variables angle and speed.

Patterns on a graphshow relationships

A good way to show a relationship between two variables is to use a graph. A graph shows one variable on the vertical (or y) axis and the second variable on the horizontal (or x) axis. Each axis is marked with the range of values the variable has. In Figure 4.12, the x-axis (angle) has values between 0 and 60 degrees. The y-axis (time) has average speed values between 0 and 300 cm/s. You can tell there is a relationship because all the points on the graph follow the same curve that slopes up and to the right. The curve tells you instantly that the average speed increases as the ramp gets steeper.

Recognizing arelationship from a

graph

The relationship between variables may be strong, weak, or there may be no relationship at all. In a strong relationship, large changes in one variable make similarly large changes in the other variable, like in Figure 4.12. In a weak relationship, large changes in one variable cause only small changes in the other. The graph on the right (below) shows a weak relationship. When there is no relationship, the graph looks like scattered dots (below left). The dots do not make an obvious pattern (a line or curve).


Chapter 4MOTION

SlopeComparing speeds You can use position vs. time graphs to quickly compare speeds. Figure 4.13

shows a position vs. time graph for two people running along a jogging path. Both runners start at the beginning of the path (the origin) at the same time. Runner A (blue) takes 100 seconds to run 600 meters. Runner B (red) takes 150 seconds to go the same distance. Runner A’s speed is 6 m/s (600 ÷ 100) and runner B’s speed is 4 m/s (600 ÷ 150). Notice that the line for runner A is steeper than the line for runner B. A steeper line on a position vs. time graph means a faster speed.

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

Calculating slope The “steepness” of a line is called its slope. The slope is the ratio of the rise (vertical change) divided by run (horizontal change). The diagram below shows how to calculate the slope of a line. Visualize a triangle with the slope as the hypotenuse. The rise is the height of the triangle. The run is the length along the base. Here, the x-axis is time and the y-axis is position. The slope of the graph is therefore the distance traveled divided by the time it takes, or the speed. The units are the units for the rise (meters) divided by the units for the run (seconds), meters per second, or m/s.

Figure 4.13: A position vs. time graph for two runners.

slope - the ratio of the rise (vertical change) to the run (horizontal change) of a line on a graph.


Chapter 4 MOTION

Figure 4.14: The position vs. time graph (top) shows the exact same motion as the speed vs. time graph (bottom).

Speed vs. time graphsConstant speed on aspeed vs. time graph

The speed vs. time graph has speed on the y-axis and time on the x-axis. The bottom graph in Figure 4.14 shows the speed vs. time for the runner. The top graph shows the position vs. time. Can you see the relationship between the two graphs? The blue runner has a speed of 5 m/s. The speed vs. time graph shows a horizontal line at 5 m/s for the entire time. On a speed vs. time graph, constant speed is shown with a straight horizontal line. At any point in time between 0 and 60 seconds the line tells you the speed is 5 m/s.

Another example The red runner’s line on the position vs. time graph has a less steep slope. That means her speed is slower. You can see this immediately on the speed vs. time graph. The red runner shows a line at 4 m/s for the whole time.

Calculating distance A speed vs. time graph can also be used to find the distance the object has traveled. Remember, distance is equal to speed multiplied by time. 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 (shown below) 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. Therefore, the area of the graph is the speed multiplied by the time. This is the distance the runner traveled.


Chapter 4MOTION

Section 4.2 Review1. On a graph of position vs. time, what do the x-values represent? What do

the y-values represent?2. Explain why time is an independent variable and position is a dependent

variable in a position versus time graph.3. What does the slope of the line on a position vs. time graph tell you about

an object’s speed?4. The graph in Figure 4.15 shows the position and time for two runners in a

race. Who has the faster speed, Robin or Joel? Explain how to answer this question without doing calculations.

5. Calculate the speed of each runner from the graph in Figure 4.15.6. The runners in Figure 4.15 are racing. Predict which runner will get to the

finish line of the race first.7. Maria walks at a constant speed of 2 m/s for 8 seconds.

a. Draw a speed vs. time graph for Maria’s motion.b. How far does she walk?

8. Which of the three graphs below corresponds to the position vs. time graph in Figure 4.16?

9. Between which times is the speed zero for the motion shown on the position vs. time graph in Figure 4.16?

Figure 4.15: Questions 4, 5, and 6.

Figure 4.16: Questions 8 and 9.


Chapter 4 MOTION

Figure 4.17: Speed vs. time graphs showing constant speed (top) and acceleration (middle and bottom).

acceleration - the rate at which velocity changes.

4.3 AccelerationConstant speed is easy to understand. However, almost nothing moves with constant speed for long. When a driver steps on the gas pedal, the speed of the car increases. When the driver brakes, the speed decreases. Even while using cruise control, the speed goes up and down as the car’s engine adjusts for hills. Another important concept in physics is acceleration. Acceleration, an important concept in physics, is how we describe changes in speed or velocity.

An example of accelerationDefinition ofacceleration

What happens if you coast down a long hill on a bicycle? At the top of the hill, you move slowly. As you go down the hill, you move faster and faster—you accelerate. Acceleration is the rate at which your speed (or velocity) changes. If your speed increases by 1 meter per second (m/s) each second, then your acceleration is 1 m/s per second.

Acceleration canchange

Your acceleration depends on the steepness of the hill. If the hill is a gradual incline, you have a small acceleration, such as 1 m/s each second. If the hill is steeper, your acceleration is greater, perhaps 2 m/s per second.

Acceleration on aspeed vs. time graph

Acceleration is easy to spot on a speed vs. time graph. If the speed changes over time then there is acceleration. Acceleration causes the line to slope up on a speed vs. time graph (Figure 4.17). The graph on the top shows constant speed. There is zero acceleration at constant speed because the speed does not change.

934.3 ACCELERATION

Chapter 4MOTION

Speed and accelerationThe difference

between speed andacceleration

Speed and acceleration are not the same thing. You can be moving (non-zero speed) and have no acceleration (think cruise control). You can also be accelerating and not moving! But if the brakes are applied and the car slows down, it is accelerating because the speed is now changing (faster to slower).

Example:Acceleration in cars

Acceleration describes how quickly speed changes. More precisely, acceleration is the change in velocity divided by the change in time. For example, suppose a powerful sports car changes its speed from zero to 60 mph in 5 seconds. In English units the acceleration is 60 mph ÷ 5 seconds = 12 mph/second. In SI units, 60 mph is about the same as 100 km/h. The acceleration is 100 km/h ÷ 5 seconds, or 20 km/h/s (Figure 4.18). A formula you can use to calculate acceleration is shown below.

Acceleration inmetric units

To calculate acceleration, you divide the change in velocity by the amount of time it takes for the change to happen. If the change in speed is in kilometers per hour, and the time is in seconds, then the acceleration is in km/h/s or kilometers per hour per second. An acceleration of 20 km/h/s means that the speed increases by 20 km/h every second.

What does “units ofseconds squared”

mean?

The time units for acceleration are often written as seconds squared or s2. For example, acceleration might be 50 meters per second squared or 50 m/s2. The steps in Figure 4.19 show how to simplify the fraction m/s/s to get m/s2. Saying seconds squared is just a math-shorthand way of talking. It is better to think about acceleration in units of speed change per second (that is, meters per second per second).

Figure 4.18: The acceleration of a sports car.

Figure 4.19: How do we get m/s2?


Chapter 4 MOTION

Figure 4.20: An acceleration example.

a. 5 m/s2

b. −2 m/s2

Solving Problems: Acceleration

A sailboat moves at 1 m/s. A strong wind increases its speed to 4 m/s in3 seconds (Figure 4.20). Calculate the acceleration.

1. Looking for: You are asked for the acceleration in meters per second.

2. Given: You are given the initial speed in m/s (v1), final speed in m/s (v2), and the time in seconds.

3. Relationships: Use the formula for acceleration:

4. Solution:

Your turn...a. Calculate the acceleration of an airplane that starts at rest and reaches a

speed of 45 m/s in 9 seconds.

b. Calculate the acceleration of a car that slows from 50 m/s to 30 m/s in 10 seconds.

2 1v vat−

=

24 m/s 1 m/s 3 m/s 1 m/s3 s 3 s

a −= = =

954.3 ACCELERATION

Chapter 4MOTION

Acceleration on motion graphsAcceleration on a

speed vs. time graphA speed vs. time graph is useful for showing how the speed of a moving object changes over time. Think about a car moving on a straight road. If the line on the graph is horizontal, then the car is moving at a constant speed (top of Figure 4.21). The upward slope in the middle graph shows increasing speed. The downward slope of the bottom graph tells you the speed is decreasing. The word acceleration is used for any change in speed, up or down.

Positive andnegative

acceleration

Acceleration can be positive or negative. Positive acceleration in one direction adds more speed each second. Things get faster. Negative acceleration in one direction subtracts some speed each second, Things get slower. People sometimes use the word deceleration to describe slowing down.

Acceleration on aposition vs. time

graph

The position vs. time graph is a curve when there is acceleration. Think about a car that is accelerating (speeding up). Its speed increases each second. That means it covers more distance each second. The position vs. time graph gets steeper each second. The opposite happens 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, becoming flat when the car is stopped.

Figure 4.21: Three examples of motion showing constant speed (top) and acceleration (middle, bottom).


Chapter 4 MOTION

Figure 4.22: A dropped ball increases its speed by 9.8 m/s each second, so its constant acceleration is 9.8 m/s2.

free fall - accelerated motion that happens when an object falls with only the force of gravity acting on it.

acceleration due to gravity - the value of 9.8 m/s2, which is the acceleration in free fall at Earth’s surface, usually represented by the small letter g.

0 1 2 3 4 5 0

10

20

30

40

50

Free Fall Speed vs. Time

Time (sec)

Sp

eed

(m/s

)

Time (sec) Speed (m/s)

0 1 2 3 4 5

0 9.8 19.6 29.4 39.2 49.0

Free fallThe definition of 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. A dropped ball is almost in free fall from the instant it leaves your hand until it reaches the ground. The “almost” is because there is a little bit of air friction that does make an additional force on the ball. A ball thrown upward is also in free fall after it leaves your hand. Even going up, the ball is in free fall because gravity is the only significant force acting on it.

The acceleration ofgravity

If air friction is ignored, objects in free fall on Earth accelerate downward, increasing their speed by 9.8 m/s every second. The value 9.8 m/s2 is called the acceleration due to gravity. The small letter g is used to represent its value. When you see the lowercase letter g in a physics question, you can substitute the value 9.8 m/s2.

Constantacceleration

The speed vs. time graph in Figure 4.22 is for a ball in free fall. Because the graph is a straight line, the speed increases by the same amount each second. This means the ball has a constant acceleration. Make sure you do not confuse constant speed with constant acceleration! Constant acceleration means an object’s speed changes by the same amount each second.

974.3 ACCELERATION

Chapter 4MOTION

Acceleration and directionA change indirection is

acceleration

If an object’s acceleration is zero, the object can only move at a constant speed in a straight line (or be stopped). A car driving around a curve at a constant speed is accelerating (in the “physics sense”) because its direction is changing (Figure 4.23). Acceleration occurs whenever there is a change in speed, direction, or both.

What “change indirection” means

What do we mean by “change in direction”? Consider a car traveling east. Its velocity is drawn as an arrow pointing east. Now suppose the car turns southward a little. Its velocity vector has a new direction.

Drawing vectors When drawing velocity arrows, the length represents the speed. A 2 cm arrow stands for 10 m/s (22 mph). A 4 cm arrow is 20 m/s, and so on. At this scale, each centimeter stands for 5 m/s. You can now find the change in velocity by measuring the length of the arrow that goes from the old velocity vector to the new one.

Turns are caused bysideways

accelerations

The small red arrow in the graphic above represents the difference in velocity before and after the turn. The change vector is 1 centimeter long, which equals 5 m/s. Notice the speed is the same before and after the turn! However, the change in direction is a sideways change of velocity. This change is caused by a sideways acceleration.

Figure 4.23: A car can change its velocity by speeding up, slowing down, or turning. The car is accelerating in each of these cases.


Chapter 4 MOTION

Figure 4.24: A soccer ball in the air is a projectile. The path of the ball is a bowl-shaped curve called a parabola.

projectile - an object moving through space and affected only by gravity.

Curved motionAcceleration and

curved motionLike velocity, acceleration has direction and is a vector. Curved motion is caused by sideways accelerations. Sideways accelerations cause velocity to change direction, which results in turning. Turns create curved motion.

An example ofcurved motion

As an example of curved motion, imagine a soccer ball kicked into the air. The ball starts with a velocity vector at an upward angle (Figure 4.24). The acceleration of gravity bends the velocity vector more toward the ground during each second the ball is in the air. Therefore, gravity accelerates the ball downward as it moves through the air. Near the end of the motion, the ball’s velocity vector is angled down toward the ground. The path of the ball makes a bowl-shaped curve called a parabola.

Projectiles A soccer ball is an example of a projectile. A projectile is an object moving under the influence of only gravity. The action of gravity is to constantly turn the velocity vector more and more downward. Flying objects such as airplanes and birds are not projectiles, because they are affected by forces generated from their own power.

Circular motion Circular motion is another type of curved motion. An object in circular motion has a velocity vector that constantly changes direction. Imagine whirling a ball around your head on a string. You have to pull the string to keep the ball moving in a circle. Your pull accelerates the ball toward you. That acceleration is what bends the ball’s velocity into a circle with you at the center. Circular motion always has an acceleration that points toward the center of the circle. In fact, the direction of the acceleration changes constantly so it always stays pointed toward the center of the circle.

994.3 ACCELERATION

Chapter 4MOTION

Section 4.3 Review1. Nearly all physics problems will use the unit m/s2 for acceleration.

Explain why the seconds are squared. Why isn’t the unit given as m/s, as it is for speed?

2. Suppose you are moving left (negative) with a velocity of -10 m/s. What happens to your speed if you have a negative acceleration? Do you speed up or slow down?

3. A rabbit starts from a resting position and moves at 6 m/s after 3 seconds. What is the acceleration of the rabbit? (Figure 4.25)

4. You are running a race and you speed up from 3 m/s to 5 m/s in 4 seconds. a. What is your change in speed?b. What is your acceleration?

5. Does a car accelerate when it goes around a corner at a constant speed? Explain your answer.

6. A sailboat increases its speed from 1 m/s to 4 m/s in 3 seconds. What will the speed of the sailboat be at 6 seconds if the acceleration stays the same? (Figure 4.26)

7. The graph at the right shows the speed of a person riding a bicycle through a city. Which point (A, B, or C) on the graph is a place where the bicycle has speed but no acceleration? How do you know?

8. What happens to the speed of an object that is dropped in free fall?

9. A ball is in free fall after being dropped. What will the speed of the ball be after 2 seconds of free fall?

10. What happens when velocity and acceleration are not in the same direction? What kind of motion occurs?

11. The Earth moves in a nearly perfect circle around the Sun. Assume the speed stays constant. Is the Earth accelerating or not?

Figure 4.25: Question 3.

Figure 4.26: Question 6.

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April 22, 2007—A young harp seal was found stranded on a beach inVirginia’s Chincoteague National Wildlife Refuge. He appeared slightly thinwith some superficial injuries. Park rangers, optimistic that he would healon his own, placed him under observation. Unfortunately, park visitorsdidn’t heed requests to keep a respectful distance from the seal.

For the seal’s and the public’s safety, he was captured and sent tothe Virginia Aquarium Stranding Center. Veterinarians treated himwith antibiotics, and soon he was consuming 10 pounds of herring aday. In less than a month, the seal grew from 35 to 66 pounds.

During that time, a 13-year-old girl asked her birthday party gueststo bring donations to the Aquarium’s Stranding Response Programinstead of gifts. With the money she collected, the aquariumpurchased a satellite tag to track the seal’s movements.

On May 19, 2007, the tag was attached and the healthy seal wasreleased back to the ocean.

What Is a Satellite Tag?A satellite tag is a palm-sized, salt-water-resistantdata collector with anantenna attached. It isglued to the fur of a seal’supper back, where itremains until the sealmolts and the tag falls off.

The tag records information including the time, date, dive depth,dive duration, and amount of time at the surface over the last sixhours. When the seal surfaces, the tag transmits this data tosatellites orbiting Earth. Sometimes there are no satellites overheadwhen the animal surfaces, so data isn’t received every day.

When data is received, instruments on the satellite record thelocation of the tag and relay the data to processing computers backon Earth. Organizations such as WhaleNet (� Internet keywordsearch: whalenet) make this information available online, whereit is used by marine scientists, government and conservationorganizations, and students.

The Seal’s Journey: Position, Time, and SpeedWhaleNet’s Satellite Tagging Observation Program (STOP) providedthe following information about the seal’s journey.

Date Time (GMT) Time Elapsed Latitude Longitude Distance Traveled since previous from previous point

point (h:min)

05/19/07 10:06 0 36.850 N 76.283 W 0 km ` (This is the release

location–First LandingState Park, Virginia).

05/30/07 04:45 258:39 42.195 N 65.554 W 1096 km

06/03/07 07:27 98:42 44.317 N 63.137 W 307 km

06/05/07 19:20 59:53 45.294 N 60.812 W 214 km

06/11/07 03:11 127:51 45.749 N 59.440 W 119 km

06/16/07 20:16 137:05 47.669 N 58.009 W 240 km

06/19/07 08:11 59:55 46.594 N 56.125 W 186 km

06/25/07 13:17 149:06 48.523 N 51.069 W 437 km

06/28/07 06:25 65:08 50.412 N 51.192 W 210 km

07/03/07 08:46 122:21 54.127 N 54.070 W 458 km

07/05/07 00:40 39:54 54.889 N 55.558 W 128 km

07/09/07 19:08 114:28 56.665 N 59.970 W 340 km

This information can be used to determine the seal’s average speedon each leg of his journey. To calculate his average speed on the first leg:

1. Convert elapsed time from h:min to hours. 258 hours 39 minutes = 258 39/60 hours = 258.65 hours

2. Plug the values into the speed formula: speed = distance / time.Speed = 1096 km / 258.65 h = 4.237 km/h

With the satellite tag attached to his back, the seal moves

toward the ocean.

High Tech

AnimalTrackers

Sea ice formed late and broke up earlyfor seven of the eleven years between1996 and 2007. Satellite tagging datahelps us monitor how animals respondto these changing conditions. Someseals travel further north. Others havetried to adapt to new habitats—forexample, seals have given birth onland instead of ice. There the pupsface new predators like foxes, wolves,and domestic and wild dogs—animalsthat don’t hunt on ice.

Marine scientists share informationabout seal population activity withgovernment agencies that monitor seal hunting and fishing industries. If the seal population declines, newregulations could be enacted to restricthunts and/or protect the seal’s foodsources and critical habitat areas, whileareas with abundant resources can beopened to the fishing industry. Themore we learn about how animalsinteract with their environments, thebetter decisions we can make abouthow we as humans use the oceans.

New Insights, Improved CoexistenceKnowing the seal’s average speed atvarious points on his journey can helpus gain insight into his behavior. Forexample, between June 5 and June 11,his average speed slowed significantly.During that time, he remained in asmall area just off the coast of CapeBreton Island. The satellite datasuggests that this area may be a“critical habitat” for the harp seal. What was he doing there? Resting?Feeding? Finding answers to thesequestions can help us make betterdecisions about how and when wehumans use this coastal region.

J. Michael Williamson, WhaleNet’sfounder and director, explains, “Similardata from tagging right whales has ledto changes in shipping lanes aroundthe whale’s feeding areas and slowedshipping traffic through areas wherewhale calves are born. Satellite taggingresearch studies have led to many newlaws and guidelines governing humanactivities around endangered species.”

What’s Nice about Sea Ice?Satellite tagging data can help us understand more about howanimals adapt to changes in their environment. For example, marinescientists are paying careful attention to how far up the Davis Straitharp seals travel. Harp seals stop their northward journey when theyrun into sea ice, rather than swimming under it, since they need tobreathe air like we do.

Harp seals rest, mate, molt, and grow new coats on the sea ice. Theyalso give birth and nurse their pups on the ice. If the ice breaks upbefore the pups are weaned, the pups may drown or be crushedbetween large chunks of ice.

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Questions:

1. What was the seal’s average speed between June 5 and June 11, 2007?

2. Name two ways satellite tagging can help humans make better decisions about how we use the oceans.

3. Research: Using an Internet keyword search for WhaleNet, find out what marine animal species are currently tagged. Use the website resources to create your own map of one animal’s journey. Compare your animal’s top speed to the harp seal’s. What questions do you have about your animal’s travels?

Satellite Tracking Map of the seal’s journey.


Chapter 4 MOTION

Chapter 4 Assessment

VocabularySelect the correct term to complete the sentences.

Section 4.1

1. Speed with direction is called ____.

2. A variable that is described using both a number and adirection is called a ____.

3. The ____ is the place where position equals zero.

4. The ____ of an object is given relative to an origin.

5. The formula for ____ is distance divided by time.

6. ____ is speed that does not change over time and ____ is thetotal distance divided by the total time of a trip.

7. The ____ of the origin of a graph are (0, 0).

8. The x-____ is horizontal on a graph.

Section 4.2

9. A mathematical diagram using two axes to represent therelationship between variables is a(n) ____.

10. The ____ of a line is the ratio of rise to run.

11. The variable usually represented on the x-axis of a graph isthe ____.

12. The variable usually represented on the y-axis of a graph isthe ____.

Section 4.3

13. The rate at which velocity changes is defined as ____.

14. An object moving in a curved path and affected only bygravity is called a(n) ____.

15. An object accelerating under only the force of gravity is saidto be in ____.

16. An object in free fall will accelerate toward Earth at9.8 m/s2, the ____.

ConceptsSection 4.1

1. What is the speed of an object that is standing still?

2. Name three common units for measuring speed.

3. Write the form of the speed equation that you would see ineach of the following scenarios.Let v = speed, t = time, and d = distance.

a. You know distance and speed and want to find the time.

b. You know time and distance and want to find the speed.

c. You know speed and time and want to find the distance.

4. How are the variables speed and velocity different? How arethey similar?

position

average speed

graph

independent variable

dependent variable

acceleration due to gravity

projectile

speed

velocity

coordinates

constant speed

acceleration

origin

vector

axis

slope

free fall

103CHAPTER 4 ASSESSMENT

Chapter 4MOTION

5. Are the following directions usually are considered positiveor negative? Write + for positive or − for negative.

a. ____ up e.____ northb. ____ down f.____ southc. ____ left g.____ east

d. ____ right h.____ west

6. If you are given x-y axes coordinates of (4, 9), which axis isrepresented by the number 9?

Section 4.2

7. You do an experiment to find out how much light is neededto make house plants grow taller. The two variables in thisexperiment are amount of light and the height of the plants.Which variable is the dependent variable and which is theindependent variable? Explain your answer.

8. Look at the graph below and answer the following questions.

a. What is the speed of runner B at 100 seconds?b. For how many seconds has runner A run at the 300-

meter position?c. Make a sketch of

this graph in your notebook. Add a line to the graph that represents a third runner who has a speed that is slower than the speeds of runner A and B. This new line should begin at the origin of the graph.

9. Which of the graphs below shows an object that is stopped?

10. Which of the graphs above shows an object moving at aconstant speed?

Section 4.3

11. How would it be possible for an object to be traveling withconstant speed and still be accelerating?

12. Can an object have a speed of zero while it has anacceleration that is not zero? Explain.

13. Which of these graphs show acceleration occurring

ProblemsSection 4.1

1. Your starting place on a track is 30 centimeters. What isyour new position if you move 10 centimeters to the left ofthis position?

2. A high-speed train travels at 300 km/h. How long (in hours)would it take the train to travel 1,500 km at this speed?


Chapter 4 MOTION

3. Lance Armstrong’s teammate, George Hincapie, averaged aspeed of 33.6 km/h in the 15th stage of the Tour de France,which took 4.00 hours. How far (in kilometers) did he travelin the race?

4. It takes Brooke 10 minutes to run 1 mile. What is her speedin miles per minute?

5. You are traveling on the interstate highway at a speed of65 mph. What is your speed in km/h? The conversion factoris: 1.0 mph = 1.6 km/h.

6. Use the speed equation to complete the following chart.

7. A pelican flies at a speed of 52 km/h for 0.25 hours. Howmany miles does the pelican travel? The conversion factor is:1.6 km/h = 1.0 mph.

8. A snail crawls 300 cm in 1 hour. Calculate the snail’s speedin each of the following units.

a. centimeters per hour (cm/h)

b. centimeters per minute (cm/min)

c. meters per hour (m/h)

9. If it takes 500 seconds for the light from the Sun to reachEarth, what is the distance to the Sun in meters? (The speedof light is 300,000,000 meters/second.)

10. Look at the graph below and give the coordinates for eachpoint.

11. A train travels 50 km/h south for 2 hours. Then the traintravels north at 75 km/h for 5 hours. Where is the train nowrelative to its starting position?

12. You want to arrive at your friend’s house by 5 p.m. Herhouse is 240 kilometers away. If your average speed will be80 km/h on the trip, when do you need to leave your house inorder to get to her house in time?

13. Starting from school, you bicycle 2 km north, then 6 kmeast, then 2 km south.

a. How far did you cycle?b. What is your final position compared to your school?c. How far and in what direction must you travel to return

to school?

14. If you walk 8 blocks north and then 3 blocks south from yourhome, what is your position compared to your home? Whatdistance did you walk?

15. You use an x-y plane to represent your position. Starting at(+150 m, −50 m), you walk 20 meters west and 30 metersnorth. What are your new coordinates?

distance (m) speed (m/s) time (s)10 6

45 5100 2

105CHAPTER 4 ASSESSMENT

Chapter 4MOTION

16. A bird flies from its nest going north for 2 hours at a speed of20 km/h and then goes west for 3 hours at 15 km/h. Whatare the distance coordinates for the bird relative to its nest?

Section 4.2

17. Draw the position vs. time graph for a person walking at aconstant speed of 1 m/s for 10 seconds. On the same axes,draw the graph for a person running at a constant speed of4 m/s.

18. Calculate the speed represented by each position vs. timegraph below.

19. Draw the speed vs. time graph that shows the same motionas each position vs. time graph above.

Section 4.3

20. A loaded garbage truck has low acceleration. It takes10 seconds to go from 0 km/h to 100 km/h. What is theacceleration of the garbage truck? How much slower is theacceleration of the garbage truck compared to theacceleration of the sports car in Figure 4.18?

21. When a ball is first dropped off a cliff in free fall, it has anacceleration of 9.8 m/s2. What is its acceleration as it getscloser to the ground? Assume no air friction.

22. Why is the position vs. time graph for an object in free fall acurve?

23. Draw a speed vs. time graph for an object accelerating fromrest at 2 m/s2.

24. Draw a speed vs. time graph for a car that starts at rest andsteadily accelerates until it is moving at 40 m/s after20 seconds. Then answer the following questions.

a. What is the car’s acceleration?b. What distance did the car travel during the 20 seconds?

25. Draw a speed vs. time graph for each of the followingsituations.

a. A person walks along a trail at a constant speed.b. A ball is rolling up a hill and gradually slows down. c. A car starts out at rest at a red light and gradually

speeds up.

Applying Your KnowledgeSection 4.1

1. If you take a one hour drive at an average speed of 65 mph,is it possible for another car with an average speed of55 mph to pass you? Explain your answer.

2. Make up your own problem! You want to end up 3 meterssouth of a starting point. Write a 5-step velocity vectorproblem that will get you to this point. You must travel in atleast three directions before you get to your end point.

2

4

6

8

00

1 2 3 4

Pos

ition

(m)

Time (s)

2

4

6

8

00

1 2 3 4

Pos

ition

(m)

Time (s)


Chapter 4 MOTION

3. Answer the following questions.

a. A herd of wild animals moves in the following directionsfrom a starting point in search of water: 10 km north, 3km east, 7 km west, 20 km south, and 4 km east. Wheredoes the herd end up relative to its starting point?

b. A watering hole is 2 km west and 2 km south of thestarting point. Does the herd make it to the wateringhole? If not, write down the directions the herd wouldneed to follow to get to the watering hole from their endposition.

Section 4.2

4. Oliver is warming up for a track meet. First he walks 1 m/sfor 100 seconds. Then he runs at 3 m/s for 200 seconds. Hisshoe comes untied, so he stops for 20 seconds to tie it.Finally he runs at 4 m/s for 200 seconds.

a. Draw a position vs. time graph of Oliver’s motion. Hint:Use the table below to calculate Oliver’s position duringeach part during his warm up.

b. Draw a speed vs. time graph of Oliver’s motion.

c. What is the total distance that Oliver travels?

d. What is Oliver's average speed during his 520 secondwarm-up?

Section 4.3

5. Look at the graph below and make up a story involvingmotion that would create a graph shaped like the one below.

6. Now draw a speed vs. time graph that shows the samemotion as the position vs. time graph above.

speed (m/s) x time (s) = position (m)

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