linear motion chapter 2. scalars vs. vectors scalars are quantities that have a magnitude, or...

Linear Motion

Chapter 2

Scalars vs. Vectors

• Scalars are quantities that have a magnitude, or numeric value which represents a size i.e. 14m or 76mph.

• Vectors are quantities which have a magnitude and a direction, for instance 12m to the right or 32mph east.

Distance vs. Displacement

Distance vs. Displacement

• The person, according to a pedometer has walked a total of 12m. That is the distance traveled.

• The person walking starts where she stops, so her displacement is zero.

Example #1

If you walk 100 m east and then turn around and walk 20 m west,

A) What is your distance you walked?

B) What is your displacement?

Distance vs. Displacement

Start

End

6m

3m

3m

1m

Distance-Add all the distances together, totals 13m.

Add the left/right pieces and the up/down pieces and use the Pythagorean Theorem.

Displacement-Measured from beginning to end.

Example #2

Start

End

6m

3m

3m

1m

6m right + 3m left=3m right

3m down + 1m down=4m down

The total displacement is 5m.

You also need to include a direction, but we will take care of that in the next chapter.

Measuring how fast you are going

• Speedv• Scalar• Standard unit is m/s

• Velocityv• Vector• Standard unit is m/s,

plus direction

t

d

time

ntdisplacemev

t

d

time

distancev

Example #3

Brad and Angelina go for a walk at 1.3 m/sec East for 30 min.

A) How far did they go?

B) Upon returning home, what distance did they travel?

C) What is their displacement?

Example #4

• A boy takes a road trip from Philadelphia to Pittsburgh. The distance between the two cities is 300km. He travels the first 100km at a speed of 35m/s and the last 200km at 40m/s. What is his average speed?

Example #4

• A boy takes a road trip from Philadelphia to Pittsburgh. The distance between the two cities is 300km. He travels the first 100km at a speed of 35m/s and the last 200km at 40m/s. What is his average speed?

Different types of velocity and speed

• Average velocity/speed• A value summarizing

the average of the entire trip.

• All that’s needed is total displacement/distance and total time.

• Instantaneous velocity• A value that

summarizes the velocity or speed of something at a given instant in time.

• What the speedometer in your car reads.

• Can change from moment to moment.

Displacement (Position) vs. Time Graphs

• Position, or displacement can be determined simply by reading the graph.

• Velocity is determined by the slope of the graph (slope equation will give units of m/s).

• If looking for a slope at a specific point (i.e. 4s) determine the slope of the entire line pointing in the same direction. That will be the same as the slope of a specific point.

• What is the position of the object at 7s?

• What is the displacement of the object from 3s to 6s?

• What is the velocity at 2s?

Class Example #1

1 2 3 4 5 6 time (s)

15

10

5

0

-5

-10

-15

Class Example #2

5 10 15 20 25 30 time (sec)

30

20

10

0

-10

-20

-30

Acceleration, a

Acceleration

t

vv

t

vonacceleratia if

• Change in velocity over time.

• Either hitting the gas or hitting the brake counts as acceleration.

• Units are m/s2

delta.• Means “change in”

and is calculated by subtracting the initial value from the final value. atvv if

Signs

• In order to differentiate between directions, we will use different signs.

• In general, it doesn’t matter which direction is positive and which is negative as long as they are consistent. However

• Most of the time people make right positive and left negative. Similarly, people usually make up positive and down negative.

• If velocity and acceleration have the same sign, the object is speeding up. If they have opposite signs, the object is slowing down.

Velocity vs. Time Graphs

• Velocity is determined by reading the graph.

• Acceleration is determined by reading the slope of the graph (slope equation will give units of m/s2).

Velocity vs. Time Graphs• Displacement is found using

area between the curve and the x axis. This area is referred to as the area under the curve (finding area will yield units of m).

• Areas above the x axis are considered positive. Those underneath the x axis are considered negative.

• Break areas into triangles (A=1/2bh), rectangles (A=bh), and trapezoids (A=1/2[b1+ b2]h).

Class Example #3

1 2 3 4 5 6 time (sec)

3

2

1

0

-1

-2

-3

Class Example #4

2 4 6 8 10 12 t (sec)

15

10

5

0

-5

-10

-15

Using linear motion equations

• We always assume that acceleration is constant.

• We use vector quantities, not scalar quantities.• We always use instantaneous velocities, not

average velocities (unless specifically stated)• Direction of a vector is indicated by sign.

Incorrect use of signs will result in incorrect answers.

Example #5

A car going 15m/s accelerates at 5m/s2 for 3.8s. How fast is it going at the end of the acceleration?

First step is identifying the variables in the equation and listing them.

Example #5

A car going 15m/s accelerates at 5m/s2 for 3.8s. How fast is it going at the end of the acceleration?

t=3.8s

vi=15m/s

a=5m/s2

vf=?

Example #6

• A penguin slides down a glacier starting from rest, and accelerates at a rate of 7.6m/s2. If it reaches the bottom of the hill going 15m/s, how long does it take to get to the bottom?

Equation for displacement

t

dv

fi vvv 21

tvd

tvvd fi 21

Example #7A racing car reaches a speed of 42 m/sec. It then begins to slow down using a parachute and braking system. It comes to rest 5.5 sec later.

A) Find how far the car moves while stopping.

B) What is the acceleration?

Equation that doesn’t require vf

tvvd fi 21 atvv if

tatvvd ii 21

)2(21 atvtd i

2

21 attvd i

Example #8How long does it take a car to cross a 30 m wide intersection after the light turns green assuming that it accelerates from rest at a constant 2.1 m/sec2?

An equation not needing t tvvd fi 2

1atvv if

atvv if

ta

vv if

a

vvvvd iffi2

1

a

vvd if

22

21

222 if vvad advv if 222

Example #9A 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?

The Big 4

atvv if advv if 222

tvatd i 2

21 tvvd fi 2

1

Example #10You 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?

Acceleration due to gravity

Gravity• Gravity causes an acceleration.• All objects have the same acceleration due

to gravity.• Differences in falling speed/acceleration

are due to air resistance, not differences in gravity.

• g=-9.8m/s2 (what does the sign mean?)• When analyzing a falling object, consider

final velocity before the object hits the grounds.

Example #11A) 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?

Hidden Variables

• Objects falling through space can be assumed to accelerate at a rate of –9.8m/s2.

• Starting from rest corresponds to a vi=0

• A change in direction indicates that at some point v=0.

• Dropped objects have no initial velocity.

Example #12Some nut is standing on the 8th street bridge in Allentown throwing rocks 6 m/sec straight down onto passing cars. If it takes 1.63 sec to hit a car,

A) how high is the bridge?

B) How fast is the rock moving just before it hits the car?

Example #13A ball is thrown up into the air at 11.2

m/sec.

A) What is the velocity at the top?

B) How high does it go?

C) How fast is it moving when it reaches its initial position?

D) How long is it in the air?

E) what is the acceleration at the top?

Homework

• Problems

Required:3, 9, 10, 12, 13, 17, 20, 22, 28, 30, 31, 33, 34, 38, 41, 45, 47, 49, 54

Additional:11, 14, 23-26, 32, 39, 42

• Graph Practice Sheet