Chapter 2 Linear MotionIn one Dimension

Motion

• When position changes over a time interval; movement; change in location

• The simplest form is when movement is in only 1 direction or line—one dimensional

• “Direction” can be up, down, left, right, positive, negative, north, south, east, west, …

Motion

• Frame of reference - the coordinate system for specifying the location of objects in space. This can be thought of as the x-axis (or y-axis) for one-dimensional motion. The choice of a frame of reference can make doing problems easier or harder

• We will typically call this x direction for this chapter

Example—position using “x”

Displacement• The length of the straight line drawn from

an object's initial position to its final position (straight line change in position)

• Displacements will be positive or negative since we are only using one dimension

• Displacement is NOT distance, it includes the direction (+ or - sign)

• Distance is total length moved, there is no positive or negative movements

Distance versus Displacement

xi

xf

Displacement• Calculating displacement:

Δx is the displacement

– Subscript i means “initial” or start position (xi)

– Subscript f means “final” or end position (xf)– Greek delta () means “change in” or

difference, which will always be final - initial

• Packet examples

f ix x x

Displacement Example

If you walk 100 m East and then turn around and walk 20 m West,

A) What is the distance you walked?

B) What is your displacement?

• A) 120 m• B) 80 m East

Velocity• Velocity includes both speed and direction• Velocity is NOT the same as speed because

velocity includes direction (velocity can be + or -), speed is only “how fast”, or magnitude

• Velocity is measured in (S.I.) units of meter per second (m/s) and is represented by the

letter “v”• There are 2 types of Velocity: average and

instantaneous (at a moment in time)

Velocity• Average Velocity is displacement divided by the

time interval during which it happened

• Speed is usually considered to be distance time, or velocity without direction (Packet ex).

• Speed is also the magnitude of a velocity, which means the “number” without direction.

timeinterval

changein position displacementaveragevelocity

changein time

f iavg

f i

x xxv v

t t t

Examples

• If you walk to down the hall (West) 75 meters in 50 seconds, what is your average velocity?

• Joe Schmo and Elliot Schmuck go for a walk at 1.3 m/sec East for 30 min.– A) How far did they go?– B) Upon returning home, how far did they travel?– C) What is their displacement?

Holt Book Problem

• Page 43, #3• An athlete swims from the north end to

the south end of a 50.0 m pool in 20.0 s and makes the return trip to the starting position in 22.0 s. What is the average velocity for:

a) the first half of the swim?b) the second half of the swim?c) the roundtrip?

Velocity• Instantaneous velocity is the velocity of

an object at one particular point in time (or one point in its path).

• We will use graphs to help understand both types of velocity

• Two types of graphs for motion study

Velocity and d vs t graphs

• Position (or displacement, d) versus time graphs provide information on velocity

• An object moving at a constant speed is represented by a straight-line on the position versus time graph. In this instance the average velocity equals a constant velocity. P

osition

Time

= Constantly changing position

Velocity and d vs t graphs

• Average velocity can be determined by drawing a straight line between any 2 points. The slope of the line gives the average velocity between those 2 points.

• Instantaneous velocity is the slope of the tangent line at a point.

• What does “-” slope mean?• What does slope = 0 mean?

Position

Time

Velocity and d vs t graphs

• Slope = rise/run

• “y” axis (rise) is position, d, or x from our equations

• “x” axis (run) is time, t same!

• So v is the slope of the position graph

f i

f i

x xv

t t

2 1

2 1

y ym

x x

Example A) Where is the particle at t =

4 sec?B) What is the total

displacement of the object over the entire trip? From t = [2, 7]sec?

C) At what interval is the object not moving?

D) How fast is the object moving at t = 3 sec? from t = [7, 8]sec?

E) What interval(s) does the object have the greatest velocity?

Changing Velocity?

• What happens on the d vs t graph if the speed is changing?

• Velocity is slope, so the slope changes• Graph is “curved”• What’s happening is called

acceleration

Acceleration• Acceleration measures the rate of change

in velocity of an object, or “how fast velocity is changing”. Any change in velocity is acceleration (speed or direction)

• Acceleration, like velocity, has a magnitude (size) and direction, so it can be + or -.

• Acceleration is measured in (S.I.) units of

meter/second/second, or m/s2 and is

represented by the letter “a”

Acceleration• The equation for average acceleration is:

• We will use constant acceleration, so “average” is the same as constant a

• The squared second:

a = v/t so in units a =

f iavg

f i

v vv changeinvelocitya

t t t time for change

2

/m s m

s s

Acceleration – 0 +

• Displacement + or –• Velocity + or – • What does + or – acceleration mean?

If + v has + a? If + v has – a? If – v has + a? If – v has – a?

Speeding up, moving positive

Slowing down, moving positive

Slowing down, moving negative

Speeding up, moving negative

Acceleration Example

Acceleration

(Left) Col. John Stapp on rocket sled. (Right) Col. Stapp’s face is contorted by the stress of rapid negative acceleration. [46.2G)http://www.youtube.com/watch?v=j4JTPg_boNw

Col: http://www.youtube.com/watch?v=s4tuvOer_GI

Acceleration Example

• With an average acceleration of -0.50 m/s2, how long will it take a cyclist to bring a bicycle with an initial speed of 13.5 m/s to a complete stop? Packet

Given: Find:

va

t

Understanding check

• What are 3 different ways to accelerate?

• Speed up• Slow down• Change direction (turn)• Back to graphing…

Acceleration and d vs t graphs

• Constant acceleration on a d vs t graph shows the changing velocity

• Since velocity is slope, the slope is changing. This results in a curved graph.

• Examples• We’ll also use another

type of graph for this-

NEXT!

Position

Time

Acceleration and v vs t Graphs

• Like position versus time graphs, velocity versus time graphs give us information about the motion of an object.

• The slope on a velocity versus time graph represents acceleration.

• If the slope is constant,

the acceleration is constantVelocity

Time

f i

f i

v va

t t

2 1

2 1

y ym

x x

Acceleration and v vs t Graphs

• The “area under the curve” on a v vs t graph is the displacement x = v t

• Negative velocity has negative area

• Area always calc from plot line to horizontal axis (time)

Velocity

Time

Example A) How fast is the particle

moving at 3.5 sec? 7.5 sec?

B) How far did the particle go in the first 3 sec?

C) What is the acceleration from t = [6, 9]sec

D) What is the acceleration at 4.68 sec?

E) What is the total distance the particle traveled? Displacement?

F) What is the average velocity from t = [0,5]sec?

Graphing summary

Graph Type Slope Area

x (or d) vs t velocity ---

v vs t acceleration displacement

a vs t --- velocity

d, v, & a graphing

d, v, a Graph.ppt

Question—Match the velocity graphs with their acceleration graph

Problems of Constant Acceleration

• When acceleration is constant, the equations for x, v, t, & a can be combined into 4 equations for problem solving, called Kinematics equations.

• Derivations for each are in the book• 5 Variables you know:

x, t, vi , vf ,a• You may need to use more than one

equation to solve problems

Constant Acceleration: Kinematics

vf = vi + a·∆t ∆x = ½ (vi + vf) ∆t vf

2 = vi2 + 2a·∆x

∆x = vi ∆t + ½ a·∆t2

See Page 54

Problem solving procedure

• Identify & list knowns/givens (3)– Check units– Identify unknowns

• Select best equation(s)• Solve:

use formula(s) and solve, then substitute values

– Or, substitute values and solve• Examples

Example Problem

• Cory’s car accelerates at 0.8 m/s2. Starting from a stop sign, what is his speed after 20 m?

• How long did the above take?

Examples

• An airplane lands moving at 80 m/s and must stop in 250 m. What acceleration must the airplane have?

• John rides on Kingda Ka at Six Flags. He knows the ride starts at rest and takes 3.5s to accelerate to full speed. He also notices the track is marked 100m from the start to full speed position. John wants to know how fast the ride is going at the end of the acceleration. How fast is Kingda Ka going? http://www.sixflags.com/greatAdventure/rides/Kingdaka.aspx

• http://www.youtube.com/watch?v=HN8nv4tVFuA http://www.cbsnews.com/videos/kingda-ka-the-countrys-tallest-and-fastest-roller-coaster/

Example You are designing an airport for small planes. One kind of

airplane that might use this airfield must reach a speed before takeoff of at least 27.8 m/sec and can accelerate at 2.0 m/sec2.

A) If the runway is 150 m

long, can this plane reach

proper speed?

B) If not, what minimum

length must it be?

Examples A sprinter can go from 0 to 7 m/sec for a distance of 2 m and continue at the same speed for the rest of a 20 m sprint.

A) What is the runner’s initial acceleration?

B) How long does it take the runner to go the entire 20 m?

Next …

• If dropped at the same time from the same height, which hits the ground first? A bowling ball or a volleyball?

• It’s a matter of acceleration—constant acceleration! EVERYTHING falls with the same constant acceleration !

(ignoring air resistance)

Free Falling…

• The most common constant acceleration is due to gravity

• When in Free Fall, and object is only acted on by gravity. We ignore air resistance.

• Acceleration due to gravity is essentially constant on the surface of the earth, so it is a constant value represented by “g”

Free Fall• “g” has been measured to be 9.81 m/s2 (or

9.8 is OK too)• Acceleration has direction; gravity always

pulls down (to center of earth)• We will pick “up” as “+” and “down” as “”,

and we’ll use y (not x) for vertical motion• So, the acceleration for anything in free fall

without air resistance is:

ay = g = 9.81 m/s2 (free fall acceleration)

Free Fall: 2 cases• Dropping:

– The instant an object is dropped, vi = 0

– Velocity is maximum as it hits the ground– http://wimp.com/waterprinter/ http://wimp.com/hadthis/ http://www.wimp.com/firstman

http://www.youtube.com/watch?v=FHtvDA0W34I

• Throwing/Shooting/Launching upward: – Once an object is in the air, what is its

acceleration?– What is the velocity at the top of its flight?

• Problems can have motion broken into parts in order to solve

Free Fall

• Up and down motion is symmetric, so velocity at any point going up is the same as the velocity going down (at the same height), except its in the opposite direction

Vi Vf = Vi

+

y direction only!

V=0

QuestionA skydiver jumps out of a helicopter. A few

seconds later, another skydiver jumps out, so they both fall along the same vertical line. Ignoring air resistance, does the vertical distance between them

A) IncreaseB) DecreaseC) Stay the same

– http://wimp.com/waterprinter/

Equations in y: vertical

vf = vi + ay·∆t ∆y = ½ (vi + vf) ∆t vf

2 = vi2 + 2ay·∆y

∆y = vi ∆t + ½ ay·∆t2

Example A) How long does it take a ball to fall from the roof of a 150 m tall building?

B) How fast is it moving when it reaches the ground?

Example Some nut standing on the 8th street bridge in Allentown throws a tennis ball 6 m/sec straight down onto passing cars but misses. If it takes 1.63 sec to hit the ground,

A) how high is the bridge?

B) How fast is the ball moving just before it hits the ground?

QuestionsA ball is thrown vertically upward. While the ball is in

freefall, does the accelerationA) IncreaseB) DecreaseC) Remain constant

After a ball is thrown vertically upward and is in the air, its speed

D) IncreasesE) DecreasesF) Decreases then increasesG) Increases then decreasesH) Remains constant

Example

• Laiken tries to throw a volleyball straight up from the bottom of the stairs to Kahli at the top. Unfortunately she comes up short, as it only goes up 4.5 m. What was the initial velocity of the ball?

• How long was the ball in the air?• How fast is the ball going when it returns

to her hand (at same height as thrown)

Free Fall

• Examples• Remember direction!•

http://wimp.com/hadthis/ http://www.wimp.com/firstman/ http://www.youtube.com/watch?v=FHtvDA0W34I new jump

• http://dsc.discovery.com/tv-shows/mythbusters/videos/lunar-lunacy.htm • http://www.wimp.com/waterfallswing

• End of Chapter 2