physics intro & kinematics quantities units vectors displacement velocity acceleration...

Physics Intro & Kinematics

• Quantities

• Units

• Vectors

• Displacement

• Velocity

• Acceleration

• Kinematics

• Graphing Motion in 1-D

Some Physics QuantitiesVector - quantity with both magnitude (size) and direction

Scalar - quantity with magnitude only

Vectors:• Displacement• Velocity• Acceleration• Momentum• Force

Scalars:• Distance• Speed• Time• Mass• Energy

Mass vs. Weight

On the moon, your mass would be the same, but the magnitude of your weight would be less.

Mass• Scalar (no direction)• Measures the amount of matter in an object

Weight• Vector (points toward center of Earth)• Force of gravity on an object

Vectors

• The length of the arrow represents the magnitude (how far, how fast, how strong, etc, depending on the type of vector).

• The arrow points in the directions of the force, motion, displacement, etc. It is often specified by an angle.

Vectors are represented with arrows

42°

5 m/s

Vectors vs. Scalars

1. One of the numbers below does not fit in the group:

a. 35 ft

b. 161 mph

c. 70° F

d. 200-m, 30° East of North

e. 12 200 people

Vectors vs. ScalarsThe answer is: 200-m, 30° East of North

Why is it different? • Numbers w/ magnitude only are called

SCALARS.• Numbers w/ magnitude and direction are

called VECTORS.

Vector Example• Particle travels from A

to B along the path shown by the dotted red line– This is the distance

traveled and is a scalar• The displacement is

the solid line from A to B– displacement is

independent of path taken between two points

– displacement is a vector– Notice the arrow

indicating direction

Other Examples of Vectors

– Displacement (of 3.5 km at 20o North of East)

– Velocity (of 50 km/h due North)

– Acceleration (of 9.81 m/s2 downward)

– Force (of 10 Newtons in the +x direction)

Notation• Vectors are written as arrows.

– length describes magnitude– direction indicates the direction of vector…

• Vectors are written in bold text in your book• Conventions for written notation shown

below…

F

Adding Vectors

Case 1: Collinear Vectors

What is the ground speed of an airplane flying with an air

speed of 100 mph into a headwind of 100 mph?

Adding Collinear Vectors

When vectors are parallel: just add magnitudes and keep the direction.

Ex: 50 mph east + 40 mph east = 90 mph east

Adding Collinear Vectors

When vectors are antiparallel: just subtract smaller magnitude from

larger - use the direction of the larger.

Ex: 50 mph east + 40 mph west = 10 mph east

Adding Perpendicular Vectors

When vectors are perpendicular:• sketch the vectors in a HEAD TO

TAIL orientation • use right triangle trig to solve for

the resultant and direction.

Ex: 50 mph east + 40 mph south = ??

An Airplane flies north with an airspeed of 650 mph. If the wind is blowing east at 50 mph, what is the speed of the plane as measured from the ground?

Adding Perpendicular Vectors

R

θ

50 mph

650 mph

Examples

Ex1: Find the sum of the forces of 30 lb south and 60 lb east.

Ex2: What is the ground speed of a speed boat crossing a river of 5mph current if the boat can move 20mph in still water?

An airplane flies north with an airspeed of 575 mph.

1. If the wind is blowing 30° north of east at 50 mph, what is the speed of the plane as measured from the ground?

2. What if the wind blew south of west?

Adding Skew VectorsWhen vectors are not collinear and not

perpendicular, we must resort to drawing a scale diagram.

• Choose a scale and a indicate a compass

• Draw the vectors Head to Tail• Draw the resultant• Measure the resultant and the angle!

Ex: 50 mph east + 40 mph south = ??

Adding Skew Vectors

Measure R with a ruler and measure θ with a protractor.

R

θ

Vector Components

• Vectors are described using their components.

• Components of a vector are 2 perpendicular vectors that would add together to yield the original vector.– Components are notated using subscripts.

F

Fx

Fy

Adding Vectors by Components

A

B


A B

Transform vectors so they are head-to-tail.


A BBy

Bx

Ax

Ay

Draw components of each vector...


A BBy

BxAx

Ay

Add components as collinear vectors!


A B By

BxAx

Ay

Draw resultants in each direction...

Ry

Rx


A B

Combine components of answer using the head to tail method...

Ry

Rx

R

q

Adding Vectors Graphically

When you have many vectors, just keep repeating the process until all are includedThe resultant is still drawn from the origin of the first vector to the end of the last vector

Adding Vectors Graphically, final

Example: A car travels 3 km North, then 2 km Northeast, then 4 km West, and finally 3 km Southeast. What is the resultant displacement?

AB

C

D

R

R is ~2.4 km, 13.5° W of Nor 103.5º from +ve x-axis.

D

C

A

B

Components of a Vector• A car travels 3 km North, then 2 km Northeast, then 4

km West, and finally 3 km Southeast. What is the resultant displacement? Use the component method of vector addition.

A B

C

D

Bx

By

Dy

Dx

Ax = 0 kmBx = (2 km) cos 45º = 1.4 kmCx = -4 kmDx = (3km) cos 45º = 2.1 km

X-components

Y-components

Ay = 3 kmBy = (2 km) sin 45º = 1.4 kmCy = 0 kmDy = (3km) sin 315º = -2.1 km

x

yN

S

W E

Components of a Vector

kmkmkmRRR yx 4.2)3.2()5.0( 2222

S

Rx = Ax + Bx + Cx + Dx = 0 km + 1.4 km - 4.0 km + 2.1 km = -0.5 km

Ry = Ay + By + Cy + Dy = 3.0 km + 1.4 km + 0 km - 2.1 km = 2.3 km

E

R

x

yN

W

Rx

Ry

Magnitude:

Direction:

78-=5.0-

3.2tan=tan= 1-1-

x

y

R

Rθ

Stop. Think. Is this reasonable? NO! Off by 180º.Answer: -78º + 180° = 102°

Components of a Vector• The x-component of a vector is the

projection along the x-axis:

• The y-component of a vector is the projection along the y-axis:

cosA

xA

sinAr

yA


Use the Pythagorean Theorem and Right Triangle Trig to solve for R and θ…

R Rx

2 Ry

2

tan 1 Ry

Rx

Units

Quantity . . . Unit (symbol) • Displacement & Distance . . . meter (m)• Time . . . second (s)• Velocity & Speed . . . (m/s)• Acceleration . . . (m/s2)• Mass . . . kilogram (kg)• Momentum . . . (kg · m/s)• Force . . .Newton (N)• Energy . . . Joule (J)

Units are not the same as quantities!

SI Prefixes

pico p 10-12

nano n 10-9

micro µ 10-6

milli m 10-3

centi c 10-2

kilo k 103

mega M 106

giga G 109

tera T 1012

Little Guys Big Guys

Kinematics definitions

• Kinematics – branch of physics; study of motion

• Position (x) – where you are located• Distance (d ) – how far you have

traveled, regardless of direction • Displacement (x) – where you are in

relation to where you started

Distance vs. Displacement• You drive the path, and your odometer

goes up by 8 miles (your distance).• Your displacement is the shorter directed

distance from start to stop (green arrow).• What if you drove in a circle?

start

stop

Speed, Velocity, & Acceleration

• Speed (v) – how fast you go

• Velocity (v) – how fast and which way; the rate at which position changes

• Average speed ( v ) – distance / time

• Acceleration (a) – how fast you speed up, slow down, or change direction; the rate at which velocity changes

Speed vs. Velocity

• Speed is a scalar (how fast something is moving regardless of its direction). Ex: v = 20 mph

• Speed is the magnitude of velocity.• Velocity is a combination of speed and

direction. Ex: v = 20 mph at 15 south of west

• The symbol for speed is v.• The symbol for velocity is type written in bold: v

or hand written with an arrow: v

Speed vs. Velocity• During your 8 mi. trip, which took 15 min., your

speedometer displays your instantaneous speed, which varies throughout the trip.

• Your average speed is 32 mi/hr.• Your average velocity is 32 mi/hr in a SE

direction.• At any point in time, your velocity vector points

tangent to your path. • The faster you go, the longer your velocity vector.

AccelerationAcceleration – how fast you speed up, slow

down, or change direction; it’s the rate at which velocity changes. Two examples:

t (s) v (mph)

0 55

1 57

2 59

3 61

t (s) v (m/s)

0 34

1 31

2 28

3 25

a = +2 mph / s a = -3 m / ss = -3 m / s

2

Velocity & Acceleration Sign ChartV E L O C I T Y

ACCELERATION

+ -

+ Moving forward;

Speeding up

Moving backward;

Slowing down

-

Moving forward;

Slowing down

Moving backward;

Speeding up

Acceleration due to Gravity

9.8 m/s2

Near the surface of the Earth, all objects accelerate at the same rate (ignoring air resistance).

a = -g = -9.8 m/s2

Interpretation: Velocity decreases by 9.8 m/s each second, meaning velocity is becoming less positive or more negative. Less positive means slowing down while going up. More negative means speeding up while going down.

This acceleration vector is the same on the way up, at the top, and on the way down!

Kinematics Formula Summary

(derivations to follow)

• vf = v0 + a t

• vavg = (v0 + vf ) / 2

• x = v0 t + ½ a t

2

• vf2 – v0

2 = 2 a x

2

1

For 1-D motion with constant acceleration:

Kinematics Derivations

a = v / t (by definition)

a = (vf – v0) / t

vf = v0 + a t

vavg = (v0 + vf ) / 2 will be proven when we do graphing.

x = v t = ½ (v0 + vf) t = ½ (v0 + v0 + a t) t

x = v0 t + a t

22

1

(cont.)

Kinematics Derivations (cont.)

2

1

vf = v0 + a t t = (vf – v0) / a

x = v0 t + a t 2

x = v0 [(vf – v0) / a] + a [(vf – v0) / a] 2

vf2 – v0

2 = 2 a x

2

1

Note that the top equation is solved for t and that expression for t is substituted twice (in red) into the x equation. You should work out the algebra to prove the final result on the last line.

Sample Problems1. You’re riding a unicorn at 25 m/s and come to

a uniform stop at a red light 20 m away. What’s your acceleration?

2. A brick is dropped from 100 m up. Find its impact velocity and air time.

3. An arrow is shot straight up from a pit 12 m below ground at 38 m/s.

a. Find its max height above ground.

b. At what times is it at ground level?

Multi-step Problems

1. How fast should you throw a kumquat straight down from 40 m up so that its impact speed would be the same as a mango’s dropped from 60 m?

2. A dune buggy accelerates uniformly at

1.5 m/s2 from rest to 22 m/s. Then the brakes are applied and it stops 2.5 s later. Find the total distance traveled.

19.8 m/s

188.83 m

Answer:

Answer:

Graphing !x

t

A

B

C

A … Starts at home (origin) and goes forward slowly

B … Not moving (position remains constant as time progresses)

C … Turns around and goes in the other direction quickly, passing up home

1 – D Motion

Graphing w/ Acceleration

x

A … Start from rest south of home; increase speed gradually

B … Pass home; gradually slow to a stop (still moving north)

C … Turn around; gradually speed back up again heading south

D … Continue heading south; gradually slow to a stop near the starting point

t

A

B C

D

Tangent Lines

t

SLOPE VELOCITY

Positive Positive

Negative Negative

Zero Zero

SLOPE SPEED

Steep Fast

Gentle Slow

Flat Zero

x

On a position vs. time graph:

Increasing & Decreasing

t

x

Increasing

Decreasing


Increasing means moving forward (positive direction).

Decreasing means moving backwards (negative direction).

Concavityt

x


Concave up means positive acceleration.

Concave down means negative acceleration.

Special Points

t

x

PQ

R

Inflection Pt. P, R Change of concavity

Peak or Valley Q Turning point

Time Axis Intercept P, S Times when you are at

“home”

S

Curve Summary

t

x

Concave Up Concave Down

Increasing v > 0 a > 0 (A)

v > 0 a < 0 (B)

Decreasing

v < 0 a > 0 (D)

v < 0 a < 0 (C)

A

BC

D

All 3 Graphs

t

x

v

t

a

t

Graphing Tips

• Line up the graphs vertically.

• Draw vertical dashed lines at special points except intercepts.

• Map the slopes of the position graph onto the velocity graph.

• A red peak or valley means a blue time intercept.

t

x

v

t

Graphing TipsThe same rules apply in making an acceleration graph from a velocity graph. Just graph the slopes! Note: a positive constant slope in blue means a positive constant green segment. The steeper the blue slope, the farther the green segment is from the time axis.

a

t

v

t

Real lifeNote how the v graph is pointy and the a graph skips. In real life, the blue points would be smooth curves and the green segments would be connected. In our class, however, we’ll mainly deal with constant acceleration.

a

t

v

t

Area under a velocity graphv

t

“forward area”

“backward area”

Area above the time axis = forward (positive) displacement.

Area below the time axis = backward (negative) displacement.

Net area (above - below) = net displacement.

Total area (above + below) = total distance traveled.

Area

The areas above and below are about equal, so even though a significant distance may have been covered, the displacement is about zero, meaning the stopping point was near the starting point. The position graph shows this too.

v

t

“forward area”

“backward area”

t

x

Area units

• Imagine approximating the area under the curve with very thin rectangles.

• Each has area of height width.• The height is in m/s; width is in

seconds.• Therefore, area is in meters!

v (m/s)

t (s)

12 m/s

0.5 s

12

• The rectangles under the time axis have negative

heights, corresponding to negative displacement.

Graphs of a ball thrown straight up

x

v

a

The ball is thrown from the ground, and it lands on a ledge.

The position graph is parabolic.

The ball peaks at the parabola’s vertex.

The v graph has a slope of -9.8 m/s2.

Map out the slopes!

There is more “positive area” than negative on the v graph.

t

t

t

Graph PracticeTry making all three graphs for the following scenario:

1. Schmedrick starts out north of home. At time zero he’s driving a cement mixer south very fast at a constant speed.

2. He accidentally runs over an innocent moose crossing the road, so he slows to a stop to check on the poor moose.

3. He pauses for a while until he determines the moose is squashed flat and deader than a doornail.

4. Fleeing the scene of the crime, Schmedrick takes off again in the same direction, speeding up quickly.

5. When his conscience gets the better of him, he slows, turns around, and returns to the crash site.

Kinematics Practice

A catcher catches a 90 mph fast ball. His glove compresses 4.5 cm. How long does it take to come to a complete stop? Be mindful of your units!

2.24 ms

Answer

Uniform Acceleration

When object starts from rest and undergoes constant acceleration:

• Position is proportional to the square of time.• Position changes result in the sequence of odd

numbers.• Falling bodies exhibit this type of motion (since g

is constant).

t : 0 1 2 3 4

x = 1 x = 3 x = 5

( arbitrary units )x : 0 1 4 9 16

x = 7

Spreadsheet Problem• We’re analyzing position as a function of time, initial

velocity, and constant acceleration.• x, x, and the ratio depend on t, v0, and a.

• x is how much position changes each second.• The ratio (1, 3, 5, 7) is the ratio of the x’s.

t (s) x (m)delta x

(m) ratio

v 0

(m/s)

a

(m/s2)

0 0 0 17.3

1 8.668.66 1

2 34.6425.98 3

3 77.9443.30 5

4 138.5660.62 7

• Make a spreadsheet like this and determine what must be true about v0 and/or a in order to get this ratio of odd numbers.

• Explain your answer mathematically.

RelationshipsLet’s use the kinematics equations to answer these:

1. A mango is dropped from a height h.

a. If dropped from a height of 2 h, would the impact speed double?

b. Would the air time double when dropped from a height of 2 h ?

2. A mango is thrown down at a speed v.

a. If thrown down at 2 v from the same height, would the impact speed double?

b. Would the air time double in this case?

Relationships (cont.)

3. A rubber chicken is launched straight up at speed v from ground level. Find each of the following if the launch speed is tripled (in terms of any constants and v).

a. max height

b. hang time

c. impact speed 3 v

9 v2 / 2 g

6 v / g

Answers

physics intro & kinematics quantities units vectors displacement velocity acceleration...

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