Motion: Kinematics Chapter 2. Scalar versus Vector Quantities Scalar Quantities –Magnitude (size)...

Motion: Kinematics

Chapter 2

Scalar versus Vector Quantities

Scalar Quantities– Magnitude (size)– “55 mph”

Vector Quantities– Magnitude (size)– Direction– “55 mph, North”

Direction Matters

Example One

x = +2 m

Example Two

x = -3 m

1 m 3 m

1 m 4 m

Speed

Two Possible Situations for Everything:

1. Object moves at a constant speed.

2. Object is being “speeded up” or “slowed down” by some force (acceleration)

“What about an object that is just sitting there?”

Speed

Average Speed = distance (in meters)

time (in seconds)

vave = x

t

Speed: Example One

During a 3.00 second time interval, a runner’s position changes from x1 = 50.0 m to x2=30.5 m. What was her average speed?

(Ans: -6.50 m/s)

Speed: Example Two

A bicycle has an average speed of 5miles/hr. How far did it go in two hours?

Speed: Example Three

How long would it take a car traveling 72 km/hr to travel 60 km?

Speed: Example Four

A car is driven at an average speed of 100.0 km/hr for two hours, then driven at an average speed of 50.0 km/hr for the next hour. What was the average speed for the three-hour trip? (Ans: 83.3 km/hr)

Since the speed changed, we have to address each part of the trip separately.

1. To find the distance for the first two hours, we use: x=vt

x= 100km/hr X 2 hr = 200 km

2. To find the distance for the last hour, we use the same process:

x= 50km/hr X 1 hr = 50 km

3. To find the average velocity, we will divide the total distance by the time:

v=x/t = 250 km /3 hr = 83.3 km/hr

Velocity: Example 5

A plane flies 300.0 miles east in 2.00 hours. The plane then turns around and flies west for 100.0 miles in 1.00 hour.

a)Calculate total distance, total time, and average speed. (133 mph)

b)Calculate the average velocity. (67.0 mph)

Average vs. Instantaneous Speed

Average Speed

averaged for an entire trip

Instantaneous Speed

Speed at an infinitely short interval

Constant velocity

Changing velocity

Acceleration

Acceleration – A change in velocity– Speed (speeding up or slowing down)

– Direction (turning a bicycle)

Speeding up or slowing down

Acceleration

• Formula

a = vvfinal – vinitial

t time

• Units = m/s2

• Acceleration can be negative (slowing down, decelerating)

Acceleration: Example 1

What is the acceleration of a bicycle that moves from rest to 5 m/s in 3 seconds?


Superman is flying through space and slows down from 24,000 m/s to 20,000 m/s in 5 seconds. What is his acceleration? What is unusual about this number?

A car starts from a complete stop at a stoplight and accelerates at 5 m/s2 for 10 seconds. What is its’ final velocity?

(Ans = 50 m/s)



A car moving at 30 m/s decelerates at

–3.0 m/s2. How long will it take for it to come to a complete stop?

(ANS: t = 10 seconds)

Kinematic Equations (Constant acceleration only)

a = v – vo

t

x = xo + vot + ½ at2

a = v2 - vo2

2x

v = v + vo

2

Time-independent

Kinematics: Example 1

An airplane takes off at 27.8 m/s and can accelerate at 2.00 m/s2. How long must the runway be if the plane is to take off safely?

(Ans: 193 m)


How long does it take a car to cross a 30.0 m intersection, from rest, if it accelerates at 2.00 m/s2?

(Ans: 5.48 s)

Warm Up

• Determine the final velocity and time for the Hot Wheels car as it goes down the track.

• Calculate acceleration.

• Calculate how far the car would travel on the level in 10 seconds.

• Would it really travel this far?


A pitcher throws a baseball at 44 m/s. What is the acceleration if the ball travels through 3.5 m from the start of the pitch to the release?

(Ans: 280 m/s2)


You wish to design an airbag that can protect a driver from a head-on collision at 100 km/h. Assume the car crumples on impact a distance of 1.00 m.

a) What is the driver’s acceleration?

b) How fast does the airbag need to act? (time)


A driver is moving at 28 m/s. It takes him 0.50 seconds to react and slam on the brakes. At that time, the brakes provide an acceleration of –6.0 m/s2. Calculate the total stopping distance.

(Ans: 79 m)

Warm Up

A horse starts from rest and can reach a speed of 4.5 m/s in 5 seconds.

a)Calculate the average speed of the horse. (2.25 m/s)

b)Calculate the acceleration of the horse. (0.9 m/s2)

c)Calculate the distance the horse will travel in 35 seconds, assuming the horse keeps accelerating. (551 m)

Aristotle versus Galileo.

In the steel cage.

Falling Objects

Aristotle (Greece, ~400 B.C.)

Heavier object fall faster than lighter ones

Galileo (1600’s, Renaissance)

All objects fall at the same rate (near the surface of the Earth, in a vacuum)

Graph Galileo’s Numbers

Galileo found that an object is at the following heights over time (from the Leaning Tower)

Time(s) Distance(m)0 560.5 551.0 511.5 452.0 362.5 253.0 12

Graph the Distance versus the Time

Dis

tan

ce (

ft)

Time (seconds)

Distance vs. Time

Distance vs. Time

0

10

20

30

40

50

60

0 0.5 1 1.5 2 2.5 3

Time (s)

Dis

tanc

e (m

)

The displacement gets greater as it falls

Acceleration due to Gravity

• Can act at a distance, is constant near the surface of the earth

• g = 9.80 m/s2

• Usually set the top of the drop at zero

yo = 0

Gravity

Why don’t people fall off the earth at the South Pole?

Gravity

Do things accelerate forever?• Not in an atmosphere. • Air drag causes “terminal

velocity”• Otherwise raindrops would

hit us with deadly speed. • Terminal velocity

– Fastest a human can drop in air

– 120 miles/hr (about 60 m/s).

Gravity: Freefall

• Freefall – falling only under the acceleration of gravity

• Object was dropped, not thrown down

y = yo + vot + ½ at2

Simplifies to:

y = ½ at2

Gravity

A rock is dropped from a cliff, calculate how far it falls during the following times. Also, draw a sketch of how it falls.

Time Distance1s2s3s4s5s

Gravity

A rock is dropped from a cliff, calculate how far it falls during the following times. Also, draw a sketch of how it falls.

Time Distance1s 4.9 m2s 19.6 m3s 44.1 m4s 78.4 m5s 123 m

0 s 0 m

1 s 4.9 m

2 s 19.6 m

3 s 44.1 m

4 s 78.4 m

Gravity: Example One (A)

A rock is dropped into a well and hits the water in 2.45 seconds. How deep is the well?

(Ans: 29.4 m)

Gravity: Example One (B)

At what speed did the rock hit the bottom of the well?

(Ans: 24.1 m/s)

Gravity: Example Two

A ball is thrown off a cliff with a initial velocity of 3.00 m/s. Calculate its position and speed after 1.00 s and 2.00 s.

(Ans: 7.90 m, 12.8 m/s and 25.6 m, 22.6 m/s)

A person throws a ball into the air with an initial velocity of 15.0 m/s.

a)How high will the ball go? (Hint: use –9.80 m/s2) (Ans: 11.5 m)

b)A person throws a ball into the air with an initial velocity of 15.0 m/s. How long is it in the air? (Ans: 3.06 s)

c)What is the ball’s velocity when it returns to the boy’s hand? (Ans: -15.0 m/s)

d)Does the ball have zero acceleration at its highest point?

Gravity: Example Three (D)

What time will the ball pass a point 8.00 M above the person’s hand?

Ans: 0.69 s and 2.37 s (you need to use the quadratic formula)

The Lava Bomb

A volcano shots a blob of lava upward at 23.2 m/s.

a.Calculate the maximum height. (27.5 m)

b.Calculate the time to reach the maximum height. (2.37 s)

c.Calculate the total time in the air. (4.73 s)

Arrow example

An arrow is shot into the air and reaches a maximum height of 30.0 m.

a.Calculate the initial speed of the arrow. (24.2 m/s)

b.Calculate the total time the arrow spent in the air. (4.94 s)

c.Calculate the time(s) when the arrow is at 15.0 m. (0.73 s and 4.21 s)

A person stands on a cliff that is 15.0 m from the ground. He throws a rock at 10.0 m/s straight up, but it falls down off the cliff.

a)Calculate the maximum height. (20.1 m)

b)Calculate the time up. (1.02 s)

c)Calculate the time down (2.03 s)

d)Calculate the speed at which it will hit the ground. (19.8 m/s)

A person stands on a building that is 25.0 m from the ground. He throws a rock at 15.0 m/s straight up, but it falls down off the building.




d)Calculate the speed at which it will hit the ground. (26.7 m/s)

A person standing on the ground throws a rock at 18.0 m/s. The rock lands on top of a roof that is 10.0 m tall.




d)Calculate the speed at which it will hit the roof. (11.3 m/s)

A person standing on the ground throws a ball at 21.0 m/s. The ball lands on top of a cliff that is 15.0 m tall.




d)Calculate the speed at which it will hit the cliff. (12.1 m/s)

e)Calculate the total distance the ball travels. (30.0 m)

Graphing

• Velocity is the slope of the x vs. t graph

v = x = (x2-x1)

t (t2-t1)

• Acceleration is the slope of the v vs. t graph

a = v= (v2-v1)

t (t2-t1)

Time (s) Displacement (m)

Velocity (m/s) Acceleration (m/s2)

1 22 X

2 44

3 66

4 88

5 110

6 132

7 154

8 176

9 198

10 220

(Use instantaneous rather than average velocity)

Displacement vs. Time

0

50

100

150

200

250

0 1 2 3 4 5 6 7 8 9 10

Time (s)

Dis

pla

cem

ent

(m)

Velocity vs. Time

0

5

10

15

20

25

0 2 4 6 8 10 12

Time (s)

Vel

oci

ty (

m/s

)

Acceleration vs. Time

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

Time (s)

Acc

eler

atio

n (

m/s

2)

Time (s) Displacement (m) Velocity (m/s) Acceleration (m/s2)

0 0

0.25 0.0625

0.50 0.2500

0.75 0.5625

1.00 1.0000

1.25 1.5625

1.50 2.2500

1.75 3.0625

2.00 4.0000

2.25 5.0625

2.50 6.2500

2.75 7.5625

3.00 9.0000(Use instantaneous rather than average velocity)


0

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5 2 2.5 3 3.5

Time(s)

Dis

pla

ce

me

nt

(m)

Velocity vs. Time

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5

Time (s)

Vel

oci

ty (

m/s

)


0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5

Time (s)

Acc

eler

atio

n (

m/s

2)

GraphingGraph the following values

time(s) displacement(m)1 222 443 664 885 1106 887 668 449 2210 0


0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10 11

Time(s)

Dis

pla

cem

ent

(m)

Velocity vs. Time

-30

-20

-10

0

10

20

30

0 2 4 6 8 10 12

Time(s)

Vel

oci

ty (

m/s

)


-50

-40

-30

-20

-10

0

0 2 4 6 8 10 12

Time(s)

Acc

eler

atio

n (

m/s

2)

x-t graphs

v-t graphs

a-t graphs

Slope = v

Slope = a Area = x

Area = v

Rules for Graphs

Link to Calculus

Displacement

Velocity

Acceleration

Slope

(Derivative)

Slope

(Derivative)

Area

(Integrate)

Area

(Integrate)

Given the following distance-time graph, calculate the velocity of the vehicle.

Given the following Velocity versus Time graph:

a.Sketch an Acceleration versus Time graph (slope)

b.Calculate the displacement (distance) travelled in

a.2 seconds c. 6 seconds

b.4 seconds d. 8 seconds

c. Draw a Displacement versus Time graph. (area)

Velocity vs. Time

02468

10121416182022242628303234363840

0 1 2 3 4 5 6 7 8 9 10 11

Time (s)

Vel

oci

ty (

m/s

)


0

1

2

3

4

5

0 2 4 6 8 10 12

Time (s)

Acc

eler

atio

n (

m/s

2)

x = t2


-50

0

50

100

150

200

250

0 2 4 6 8 10 12

Time (s)

Dis

plac

emen

t (m

)

Given the following Velocity versus Time graph:

a.Sketch an Acceleration versus Time graph

b.Calculate the total distance travelled at the following times:

a.15 s

b.20 s

c. 25 s

d.30 s

Velocity vs. Time

0

5

10

15

20

0 5 10 15 20 25 30 35

Time(s)

Vel

oci

ty (

m/s

)

Given the following a-t graph, calculate the velocity of the falling object at:

a.1 sb.2sc.3 sd.4 se.5 s

Kinematic Equations

v = vo + at

x = xo + vot + ½ at2

v2 = vo2 + 2ax

v = v + vo

2

2) 36 min8 a) 10.4 m/s b) +3.5 m/s10) 4.43 h and 881 km/h14) 5.2 s16) -6.3 m/s2 (0.64 g)20) -2.4 m/s2

22) a = 4.41 m/s2, t = 2.61 s24) 33 m/s26)a = -390 m/s2, 40 g’s34) 60.0 m

34) 60.0 m

36) h = 32 m, t = 5.1 s

38) 16.2 m/s, 13 m

40) vo = 4.85 m/s, t = 0.990 s

44 a) 12.8 m/s b) 0.735s, 3.35s

c) On the way up and on the way down

62) 1.3 m

52.a) constant v is between 0 and 17 s

b) t = 28 s

c) t = 38 s

d) both directions

54 a) 2.5 m/s2 and 0.6 m/s2

b) 450 m

Answers to practice quiz

1. a) 4.86 m/s2 b) 156 m c) 48.6 m/s

2. a) 5.7 m/s b) 0.58 s c) No, g

3. a) 6.25 X 1014 m/s2 b) 7.95 X 10-9 s

c) 2.52 X 106 m/s

4. a) 1.28 mb) 0.51 s c) 5 m/s d) 11.1 m/s

5. a) 165 m/s2 b) 330 m c) 165 m/s

6. A) 99.2 m b) 44.1 m/s

7. A) 3.36 m/s2 b) 1.64 s

Answers to Graphing

1 a) 0

b) 5 m/s

c) No, distance not changing

d) Turns around

e)-5 m/s (opposite direction)

2 a) 2 m/s2 b) 0 m/s2 c) No, const. v

d) -4 m/s2 e) 400 m f) 600 m

3 a) 55 s b) 55 s onward c) 100 m/s

d) 300 m/s e) 362.5 m/s f) 162.5 m/s

4 a) Car Y at rest b) 0.5 m/s2 c) 400 m

d) 400 m e) at 40 s

5 a) Scott is moving in the positive direction, Kevin, negative

b) 8 hours

c) Kevin

d) VKevin = -0.5 cm/hr VScott = 0.1 cm/hr

Motion: Kinematics Chapter 2. Scalar versus Vector Quantities Scalar Quantities –Magnitude (size)...

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