VECTORSAND

SCALARS

SCALARS35 km long 35 mm wide

3 hours 35 minutes 35.8 kg

2000 kcal 20% efficiency

Vectors

5 km south35 m/s south-west

45 N upward9.8 m/s2 downward

Symbol for vectors:

Typed: v, a, F, x

Written:

Multiplying vectors by scalars

Vector x scalar = vector

Multiplying vectors by scalars

Comparing vectors

Which vectors have the same magnitude?

Which vectors have the same direction?

Which arrows, if any, represent the same vector?

Adding vectors

Subtracting vectors (+ negative)

Subtracting vectors (“fork”)

Check your understanding

Construct and label a diagram that shows the vector sum 2A + B. Construct and label a second diagram that shows B + 2A.

Construct and label a diagram that shows the vector sum A – B/2. Construct and label a second diagram that shows B/2 - A.

Adding vectors algebraically

Adding vectors that are

perpendicular

Example

A person walks 5 km east and 3 km south. What is the person’s total displacement?

Hint: displacement is a vector quantity, the answer should have magnitude (km) and direction (degrees)

Relative Motion• A person is sitting in a bus that is moving north

2 m/s. a)What is the passenger’s velocity relative to a person standing on the sidewalk nearby?b) What is the passenger’s velocity relative to the person sitting next to him?c) What is his speed relative to a car moving at 4 m/s in the opposite late towards the bus?

Relative motion

• A swimmer moves at 2 m/s in a pool. a)What will be her speed relative to a person on the river bank if she swims downstream and the current is 5 m/s?

b) What will her speed be if she tries to swim against the current?

• A boat is crossing a river moving south with a speed of 4.0m/s relative to the current. The current moves due east at a speed of 2.0 m/s relative to the land.

a) How far across the river will the boat be in 2 seconds?

b) How far down the river the boat be in 2 seconds?

c) What will the boat’s total displacement be (from the place where it started).

Relative motion

• A motorboat is crossing a 350-m wide river moving 10 m/s relative to water. If current is 3 m/s east, how far down the stream will the boat end up when it reaches the opposite bank? What will the boat’s total displacement be? What will the boat’s velocity be relative to water?

• (more than one way to solve!)

Adding vectors that are NOT

perpendicular

Adding vectors algebraically

Adding vectors

Adding vectors

RESOLVING VECTORS

How fast must a truck travel to stay beneath an airplane that is moving 105 km/h at an angle of 25° to the ground?

Practice finding components

• 35 km/h 20o NE• 120 m 35o WN• 3.2 m/s2 40o SW• 65 km 70o ES

X: 33 km/h Y: 12 km/hX: - 69 m Y: 99 mX: - 2.5 m/s2 Y: - 2.1 m/s2

X: 61 km Y: - 22km

In groups, on big graph paper, illustrate the following

• One of the holes on a golf course lies due east of the tee. A novice golfer flubs his tee shot so that the ball lands only 64 m directly northeast of the tee. He then slices the ball 30° south of east so that the ball lands in a sand trap 127 m away. Frustrated, the golfer then blasts the ball out of the sand trap, and the ball lands at a point 73 m away at an angle 27° north of east. At this point, the ball is on the putting green and 14.89 m due north of the hole. To his amazement, the golfer then sinks the ball with a single shot.

• Use algebraic formulas to find the x and y components of each displacement vector.

Shot 1 x component _____________ y component _____________Shot 2 x component _____________ y component _____________Shot 3 x component _____________ y component _____________Shot 4 x component _____________ y component _____________

Calculate

• Find the total displacement (to the nearest meter) the golf ball traveled from the tee to the hole. Assume the golf course is flat. (Hint: Which component of each displacement vector contributes to the total displacement of the ball between the tee and the hole?)

Adding vectors

Adding vectors algebraically

Example, p. 94, #1

• A football player runs directly down the field for 35 m before turning to the right at an angle of 25° from his original direction and running an additional 15 m before getting tackled. What is the magnitude and direction of the runner’s total displacement?

Example 2, p 94, #4

• An airplane flying parallel to the ground undergoes two consecutive displacements. The first is 75 km 30.0° west of north, and the second is 155 km 60.0° east of north. What is the total displacement of the airplane?

Answer: 171 km at 34° east of north

Table hockey lab

Projectile motion

Free-falling while moving horizontally

Moving in two directions at the same time

Projectile motion

Parabolic trajectory

Vertical motion: with acceleration of -9.8 m/s2 and initial velocity vy and vertical displacement Δy Horizontal motion: constant vx and horizontal displacement Δx

Projectile motionVertical HorizontalAccelerated Constant speed

2

2

2 2

vertical componen

9.8 /

1( )

t

2

2

f i

i

f i

tt

t

v

vg m s

v v g

y v t g

v v g y

horizontal component

i f

xv

v v v

x v

x

t

vv

t

t

An object launched horizontally

Find the initial speed of the bike

GivenEquations / Substitution Solving

Horizontal Vertical

Lab – Launching a Projectile Horizontally

• Work in pairs• Each person should have his / her set of data• Each student turns in a report with his / her

name on it.

Projectile Launched at an angle

Projectile has initial velocity in both directions

Example

• A football is kicked 22 m/s at 40o to the horizontal. Calculate the maximum height and horizontal range the football will reach before it falls down (providing no one catches it).

1st step:

2 2

2

2 ( 0)

2

f i f

i

H y

v v g y v

vy

g

?

R x

x v t

t

0 14

9.8

vg t

t

2 1.44 2.9( )t s

17 2.8 49( )R m

Example 2 (p.110, #35)

• A place kicker must kick a football from a point 36.0 m from the goal. As a result of the kick, the ball must clear the crossbar, which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 20.0 m/s at an angle of 53° to the horizontal.

• a. By how much does the ball clear or fall short of clearing the crossbar?

• b. Does the ball approach the crossbar while still rising or while falling?

Given Units Equations Substitute/Solve