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Accelerated Motion

Chapter

3

In this chapter you will

- Develop descriptions of accelerated motions.
- Use graphs and equations to solve problems

involving moving objects. - Describe the motion of objects in free fall.

Table of Contents

Chapter

3

Chapter 3 Accelerated Motion

Section 3.1 Acceleration Section 3.2 Motion

with Constant Acceleration Section 3.3 Free Fall

- Homework
- Read Chapter 3. Complete Study Guide.
- Graphical Analysis Packet
- HW 3 handout

Warmup Which Velocity Is It?

Section

3.1

Physics Daily Warmup 16

- There are two types of velocity that we encounter

in our everyday lives. Instantaneous velocity

refers to how fast something is moving at a

particular point in time, while average velocity

refers to the average speed something travels

over a given period of time. - For each use of velocity described below,

identify whether it is instantaneous velocity or

average velocity. - The speedometer on your car indicates you are
- going 65 mph.
- A race-car driver was listed as driving 120 mph
- for the entire race.
- A freely falling object has a speed of 19.6 m/s

after - 2 seconds of fall in a vacuum.
- 4. The speed limit sign says 45 mph.

instantaneous

average

instantaneous

instantaneous

Acceleration

Section

3.1

In this section you will

- Define acceleration.
- Relate velocity and acceleration to the motion of

an object. - Create velocity-time graphs.

Acceleration

Section

3.1

Changing Velocity

- You can feel a difference between uniform and

nonuniform motion.

- When you move in
- nonuniform motion, you feel pushed or pulled.
- In contrast, when you are in
- uniform motion and your eyes are closed, you

feel as though you are not moving at all.

Acceleration

Section

3.1

Changing Velocity

- Consider the motion diagrams below showing the

distance between successive positions.

Acceleration

Section

3.1

Changing Velocity

- There are two major indicators of the change in

velocity in this motion diagram. The change in

the spacing of the stick figures or dots and the

differences in the lengths of the velocity

vectors indicate the changes in velocity.

Acceleration

Section

3.1

Changing Velocity

- If an object speeds up, each subsequent velocity

vector is longer. - If the object slows down, each vector is shorter

than the previous one. - Both types of motion diagrams give an idea of how

an objects velocity is changing.

Acceleration

Section

3.1

Velocity-Time Graphs

Play ch3_1_movanim.

Acceleration

Section

3.1

Average and Instantaneous Acceleration

- The rate at which an objects velocity changes is

called the acceleration of the object. When the

velocity of an object changes at a constant rate,

it has a constant acceleration.

- The average acceleration of an object is the

change in velocity during some measurable time

interval divided by that time interval.

- Average acceleration is measured in m/s2.
- The change in velocity at an instant of time is

called instantaneous acceleration.

Acceleration

Section

3.1

Instantaneous Acceleration

- The instantaneous acceleration of an object can

be found by - drawing a tangent line on the velocity-time

graph at the point of time in which you are

interested. - The slope of this line is equal to the

instantaneous acceleration.

Example Tangent line is drawn at t 1.0 s and t

5.0 s

Acceleration

Section

3.1

Average Acceleration

- The average acceleration of an object can be

found by plotting two points which define the

time interval, connecting the points with a

straight line, ad finding the slope of the line.

Example slope of the line indicates the average

velocity between 1.0 and 5.0 s.

Section

Velocity Time Graph Example

3.1

a) How would you describe the sprinters velocity

and acceleration as shown on the graph? From the

graph, note that the sprinters velocity starts

at zero, increases rapidly for the first few

seconds, and then, after reaching about 10.0 m/s,

remains almost constant.

Section

Velocity Time Graph Example

3.1

b) What is his instantaneous acceleration at t

1 second and t 5 seconds? Instantaneous

acceleration Draw a tangent to the curve at t

1.0 s and t 6.0 s. Find the slope of each line.

Section

Velocity Time Graph Example

3.1

Solve for acceleration at 1.0 s

Solve for acceleration at 5.0 s

The acceleration is not constant because it

changes from 3.4 m/s2 to 0.03 m/s2 at 5.0 s. The

acceleration is in the direction chosen to be

positive because both values are positive.

Section

Velocity Time Graph Example

3.1

c) What is his average acceleration between 1 and

5 seconds? Average acceleration Draw a line

between the points at t 1 s and t 5 s. Find

the slope of the line.

The slope is 4 m/s 5 s 0.8 m/s2

Acceleration

Section

3.1

Positive and Negative Acceleration

- These four motion diagrams represent the four

different possible ways to move along a straight

line with constant acceleration.

- Positive direction, speeding up
- Positive direction, slowing down
- Negative direction, speeding up
- Negative direction, slowing down

Acceleration

Section

3.1

Positive and Negative Acceleration

- When the object is speeding up, the velocity and

acceleration vectors point in the same direction.

(case 1 and 3) - When the object is slowing down, the velocity and

acceleration vectors point in opposite directions

(case 2 and 4) - Both the direction of an objects velocity and

its direction of acceleration are needed to

determine whether it is speeding up or slowing

down.

Acceleration

Section

3.1

Positive and Negative Acceleration

- An object has a positive acceleration when the

acceleration vector points in the positive

direction and a negative acceleration, when the

acceleration vector points in the negative

direction. - The sign of acceleration does not indicate

whether the object is speeding up or slowing down.

Acceleration

Section

3.1

Determining Acceleration from a v-t Graph

- Velocity and acceleration information also is

contained in velocity-time graphs.

- Graphs A, B, C, D, and E, as shown on the right,

represent the motions of five different runners. - Positive velocity in this graph means the

direction is east. Negative velocity means the

direction is west.

Section

Determining Acceleration from a v-t Graph

3.1

- Describe the direction of motion, velocity and

acceleration for - Graph A
- Zero slope means zero acceleration constant

velocity towards the east. - Graph B
- Starting with zero velocity, picking up speed,

moving towards the east. The straight line

indicates constant acceleration. - Graph C
- Moving towards the east while slowing down and

eventually stopping. Slowing down with a constant

negative acceleration.

Section

Determining Acceleration from a v-t Graph

3.1

- Graph D
- Moving towards the west while slowing down, turns

around, then moves east while speeding up. - Graph E
- Moving west with constant velocity, zero

acceleration. - Remember, for a velocity-time graph
- - positive velocity occurs when the line is

anywhere above the x-axis - - positive velocity means the object is moving

in the positive direction, which might be

east, to the right, etc. - - the slope of the line indicated the

acceleration - - a straight line means constant acceleration

Acceleration

Section

3.1

Determining Acceleration from a v-t Graph

- The following equation expresses average

acceleration as the slope of the velocity-time

graph.

- Average acceleration is equal to the change in

velocity, divided by the time it takes to make

that change.

Acceleration

Section

3.1

Example Suppose you run a wind sprints back

and forth across the gym. You run at a speed of

4.0 m/s toward the wall, touch, and run back at

the same speed. The whole trip takes 10 seconds.

What is your average acceleration if the positive

direction is toward the wall?

Givens vi 4 m/s vf -4 m/s ?t 10

s Unknown a Equation Substitute and

Solve Sense The negative sign means the

acceleration is away from the wall. Change in

direction of motion results in acceleration.

Section Check

Section

3.1

Question 1

- Which of the following statements correctly

define acceleration?

- Acceleration is the rate of change of

displacement of an object. - Acceleration is the rate of change of velocity of

an object. - Acceleration is the amount of distance covered in

unit time. - Acceleration is the rate of change of speed of an

object.

Section Check

Section

3.1

Answer 1

- Answer B

Reason The rate at which an objects velocity

changes is called acceleration of the object.

Section Check

Section

3.1

Question 2

- What happens when the velocity vector and the

acceleration vector of an object in motion are in

same direction?

- The acceleration of the object increases.
- The speed of the object increases.
- The object comes to rest.
- The speed of the object decreases.

Section Check

Section

3.1

Answer 2

- Answer B

Reason When the velocity vector and the

acceleration vector of an object in motion are in

same direction, the speed of the object increases.

Section Check

Section

3.1

Question 3

- On the basis of the velocity-time graph of a car

moving up a hill, as shown on the right,

determine the average acceleration of the car?

- 0.5 m/s2
- -0.5 m/s2

- 2 m/s2
- -2 m/s2

Section Check

Section

3.1

Answer 3

- Answer B

Reason Average acceleration of an object is the

slope of the velocity-time graph.

vf 0 vi 25 m/s tf 50 s ti 0 a vf

vi 0 25 m/s - 0.5 m/s2 tf ti 50

s - 0

Section Check

Section

3.2

Practice Problems, p. 64 6, 7, 9, 10.

Motion with Constant Acceleration

Steel Ball Race, p. 58

Motion with Constant Acceleration

Section

3.2

In this section you will

- Interpret position-time graphs for motion with

constant acceleration. - Determine mathematical relationships among

position, velocity, acceleration, and time. - Apply graphical and mathematical relationships to

solve problems related to constant acceleration.

Motion with Constant Acceleration

Section

3.2

Velocity with Average Acceleration

- If an objects average acceleration during a time

interval is known, then it can be used to

determine how much the velocity changed during

that time. - The definition of average acceleration

can be rewritten as follows

Motion with Constant Acceleration

Section

3.2

Velocity with Average Acceleration

- The equation for final velocity with average

acceleration can be written as follows

- The final velocity is equal to the initial

velocity plus the product of the average

acceleration and time interval.

vi

Motion with Constant Acceleration

Section

3.2

Velocity with Average Acceleration

- In cases in which the acceleration is constant,

the average acceleration, a, is the same as the

instantaneous acceleration, a. - The equation for final velocity can be rearranged

to find the time at which an object with constant

acceleration has a given velocity. - It also can be used to calculate the initial

velocity of an object when both the velocity and

the time at which it occurred are given.

Motion with Constant Acceleration

Section

3.2

Position with Constant Acceleration

- The position data at different time intervals for

a car with constant acceleration are shown in the

table. - The data from the table are graphed as shown on

the next slide.

Motion with Constant Acceleration

Section

3.2

Position with Constant Acceleration

- The graph shows that the cars motion is not

uniform the displacements for equal time

intervals on the graph get larger and larger.

- The slope of a position-time graph of a car

moving with a constant acceleration gets steeper

as time goes on.

Motion with Constant Acceleration

Section

3.2

Position with Constant Acceleration

- The slopes from the position time graph can be

used to create a velocity-time graph as shown on

the right. - Note that the slopes shown in the position-time

graph are the same as the velocities graphed in

velocity-time graph.

Motion with Constant Acceleration

Section

3.2

Position with Constant Acceleration

- A velocity-time graph does not contain any

information about the objects position. - However, the velocity-time graph does contain

information about the objects displacement. - Recall that for an object moving at a constant

velocity,

Motion with Constant Acceleration

Section

3.2

Position with Constant Acceleration

- On the graph shown on the right, v is the height

of the plotted line above the t-axis, while ?t is

the width of the shaded rectangle. The area of

the rectangle, then, is v?t, or ?d. Thus, the

area under the v-t graph is equal to the objects

displacement. - The area under the v-t graph is equal to the

objects displacement.

Motion with Constant Acceleration

Section

3.2

Finding the Displacement from a v-t Graph Example

1

The v-t graph below shows the motion of an

airplane. Find the displacement of the airplane

at ?t 1.0 s and at ?t 2.0 s.

Motion with Constant Acceleration

Section

3.2

Finding the Displacement from a v-t Graph

The displacement is the area under the v-t graph.

The time intervals begin at t 0.0.

Motion with Constant Acceleration

Section

3.2

Finding the Displacement from a v-t Graph

Identify the given and unknown variables.

Given v 75 m/s ?t 1.0 s ?t 2.0 s

Unknown ?d ?

Motion with Constant Acceleration

Section

3.2

Finding the Displacement from a v-t Graph

Solve for displacement during ?t 1.0

s. Equation

Substitute v 75 m/s, ?t 1.0 s Solve

Motion with Constant Acceleration

Section

3.2

Finding the Displacement from a v-t Graph

Solve for displacement during ?t 2.0 s.

Substitute v 75 m/s, ?t 2.0 s

Motion with Constant Acceleration

Section

3.2

Finding the Displacement from a v-t Graph Sense

- Are the units correct?
- Displacement is measured in meters.
- Do the signs make sense?
- The positive sign agrees with the graph.
- Is the magnitude realistic?
- Moving a distance to about one football field is

reasonable for an airplane.

Motion with Constant Acceleration

Section

3.2

An Alternative Expression

- Often, it is useful to relate position, velocity,

and constant acceleration without including time. - The three equations for motion with constant

acceleration are summarized in the table.

Section Check

Section

3.2

Question 1

- A position-time graph of a bike moving with

constant acceleration is shown on the right.

Which statement is correct regarding the

displacement of the bike?

- The displacement in equal time interval is

constant. - The displacement in equal time interval

progressively increases.

- The displacement in equal time interval

progressively decreases. - The displacement in equal time interval first

increases, then after reaching a particular point

it decreases.

Section Check

Section

3.2

Answer 1

- Answer B

Reason You will see that the slope gets steeper

as time goes, which means that the displacement

in equal time interval progressively gets larger

and larger.

Section Check

Section

3.2

Question 2

- A car is moving with an initial velocity of vi

m/s. After reaching a highway, it moves with a

constant acceleration of a m/s2, what will be the

velocity (vf) of the car after traveling for t

seconds?

- vf vi at
- vf vi 2at
- vf2 vi2 2at
- vf vi at

Section Check

Section

3.2

Answer 2

- Answer A

Reason Since a ?v/?t vf - vi a (tf -

ti) Also since car is starting from rest, ti

0 Therefore vf vi at (where t is the total

time)

Section

Section Check

3.2

Question 3

If you were given initial and final velocities

and the constant acceleration of an object, and

you were asked to find the displacement, what

equation would you use?

- vf vi at
- df di vi t ½ at2
- vf2 vi2 2a(df - di)
- vf vi at

Section

Section Check

3.2

Answer 3

Answer C

Reason Kinematics equation number 3 does not

require time. You are not given time in the

problem.

Motion with Constant Acceleration

Section

3.2

Practice Problems p.65 20, 21. Section Review

p.71 34, 39.

Free Fall

Section

3.3

Free Fall

In this section you will

- Define acceleration due to gravity.
- Solve problems involving objects in free fall.

Free Fall

Section

3.3

Acceleration Due to Gravity

- free fall the motion of a body when air

resistance is negligible and the motion can be

considered due to the force of gravity alone. - After a lot of observation, Galileo concluded

that, neglecting the effect of the air, all

objects in free fall had the same acceleration. - It didnt matter what they were made of, how

much they weighed, what height they were dropped

from, or whether they were dropped or thrown. - The acceleration of falling objects, given a

special symbol, g, is equal to 9.80 m/s2. - The acceleration due to gravity is the

acceleration of an object in free fall that

results from the influence of Earths gravity.

Free Fall

Section

3.3

Acceleration Due to Gravity

ch 3_4_movanim

Free Fall

Section

3.3

Acceleration Due to Gravity

- At the top of the flight, the balls velocity is

0 m/s. What would happen if its acceleration were

also zero? Then, the balls velocity would not be

changing and would remain at 0 m/s. - If this were the case, the ball would not gain

any downward velocity and would simply hover in

the air at the top of its flight. - Because this is not the way objects tossed in the

air behave on Earth, you know that the

acceleration of an object at the top of its

flight must not be zero. Further, because you

know that the object will fall from that height,

you know that the acceleration must be downward.

Free Fall

Section

3.3

Acceleration Due to Gravity

- Amusement parks use the concept of free fall to

design rides that give the riders the sensation

of free fall. - These types of rides usually consist of three

parts the ride to the top, momentary suspension,

and the plunge downward. - When the cars are in free fall, the most massive

rider and the least massive rider will have the

same acceleration.

Free Fall

Section

3.3

Acceleration Due to Gravity

- Example Suppose the free-fall ride at an

amusement park starts at rest and is in free fall

for 1.5 s. What would be its velocity at the end

of 1.5 s? How far would it fall? - Choose a coordinate system with a positive axis

upward and the origin at the initial position of

the car. Because the car starts at rest, vi would

be equal to 0.0 m/s. - Givens vi 0.0 m/s, di 0.0 m, ti 0.0 s, tf

1.5 s, a -9.8 m/s2 - Unknown vf

Free Fall

Section

3.3

Acceleration Due to Gravity

- To calculate the final velocity, use the equation

for velocity with constant acceleration. - Equation
- Substitute Solve
- Sense Negative velocity means down units are

OK.

Free Fall

Section

3.3

Acceleration Due to Gravity

- How far does the car fall? Use the equation for

displacement when time and constant acceleration

are known. - Equation
- Substitute Solve
- Sense The displacement is negative because it

fell, and the units are meters. Looks good!

Section Check

Section

3.3

Question 1

- What is free fall?

Section Check

Section

3.3

Answer 1

- Free Fall is the motion of the body when air

resistance is negligible and the action can be

considered due to gravity alone.

Section Check

Section

3.3

Question 2

- If a stone is thrown vertically upwards with a

velocity of 25 m/s, what will be the velocity of

the stone after 1 second?

- 9.8 m/s
- 15.2 m/s
- 25 m/s
- 34.8 m/s

Section Check

Section

3.3

Answer 2

- Answer B

Reason Since the ball is thrown upwards, the

velocity and acceleration are in opposite

directions, therefore the speed of the ball

decreases. After 1 s, the balls velocity is

reduced by 9.8 m/s (as acceleration due to

gravity is 9.8 m/s2), so it is now traveling at

25 m/s 9.8 m/s 15.2 m/s.

Section Check

Section

3.3

Question 3

- If a 50-kg bag and a 100-kg bag are dropped from

a height of 50 m. Which of the following

statement is true about their acceleration?

(Neglect air resistance)

- 100-kg bag will fall with a greater acceleration.
- 50-kg bag will fall with a greater acceleration.
- Both will fall at the same and constant rate of

acceleration. - Both will fall at the same rate of acceleration,

which changes equally as time goes.

Section Check

Section

3.3

Answer 3

- Answer C

Reason Any body falling freely towards Earth,

falls with a same and constant acceleration of

9.8 m/s2. It doesnt matter how much it weighed

and what height it was dropped from.

Free Fall

Section

3.3

- Exit Ticket Please write in full sentences.
- Describe the velocity and acceleration of a ball

that is tossed in the air and comes back down.

Free Fall

Section

3.3

- Practice Problems p.74 42, 44.
- HW 3 handout.

Chapter

Physics Chapter 2 3 Test Information The test

is worth 46 points total. Matching 12

questions, 12 points total Problems 7 questions,

34 points total Know - vocabulary for both

chapters - how to interpret and draw

position-time and velocity-time graphs - how

to use the 3 kinematics equations to solve

problems - how to express answers with correct

units and sig figs

3