Physics: Problem Solving

Physics: Problem Solving

Chapter 4 Vectors

Physics: Problem Solving

Chapter 4 Vectors

Chapter 4: Vectors

Vector ReviewTrigonometry for PhysicsVector Addition—AlgebraicVector Resolution

Chapter 4: Vectors

Vector:A measurement with both magnitude and direction

Magnitude:A numerical value

Direction:+/–North, South, East, WestWhich is larger 2m/s or – 3 m/s?

Chapter 4: Vectors

Vector: What does it mean when an object has…..?Velocity—negative and positiveAcceleration—negative and positive

Chapter 4: Vectors

Vectors can be added together both graphically and algebraicallyGraphic addition: using arrows of proper length and direction to add vectors togetherAlgebraic addition: using trigonometry to add vectors together

Chapter 4: Vectors

Graphic addition: “Butt-Head” Method1. Draw first vector to scale (magnitude &

direction)2. Draw second vector to scale. Connect the

“butt” of the second vector to the “head” of the first vector

3. Repeat Step #2 until you run out of vectors.4. Draw Resultant (?). Connect the “butt” of

the first vector with the “head” of the last vector—butt-butt, head-head

Chapter 4: Vectors

Graphic addition: “Butt-Head” MethodA dog walks 15 km east and then 10

km west find the sum of the vectors (resultant)

Chapter 4: Vectors

Graphic addition: “Butt-Head” MethodA dog walks 15 km east and then 10 km west

find the sum of the vectors (resultant)1. Draw first vector to scale (magnitude &

direction)

Chapter 4: Vectors

Graphic addition: “Butt-Head” MethodA dog walks 15 km east and then 10 km west

find the sum of the vectors (resultant)2. Draw second vector to scale. Connect the

“butt” of the second vector to the “head” of the first vector

Chapter 4: Vectors

Graphic addition: “Butt-Head” MethodA dog walks 15 km east and then 10 km west

find the sum of the vectors (resultant)4. Draw Resultant (?). Connect the “butt” of

the first vector with the “head” of the last vector—butt-butt, head-head

Measure: 5 km east

Chapter 4: Vectors

Graphic addition: “Butt-Head” MethodA dog walks 15 km east and then 10

km south find the sum of the vectors (resultant)

Measure: 18 km

34 south of east

Chapter 4: Vectors

Graphic addition: “Butt-Head” MethodExamples

A shopper walks from the door of the mall to her car 250 m down a row of cars, then turns 90 to the right and walks another 60 m. What is the magnitude and direction of her displacement from the door?Answer

257 meters 13.5 from door

Chapter 4: Vectors

This presentation deals with adding vectors algebraicallyBut first…..you need to learn some trigonometry!!!

Vector Addition-Graphic

Let’s test our knowledge! (1-5)

All the Trig. you need to know for Physics (almost)

This is a right triangle

All the Trig. you need to know for Physics (almost)

This is an angle (-theta) in a right triangle

All the Trig. you need to know for Physics (almost)

This is the hypotenuse (H) of a right triangle

All the Trig. you need to know for Physics (almost)

This is the side adjacent (A) to the angle (-theta) in a right triangle

All the Trig. you need to know for Physics (almost)

This is the side opposite (O) the angle (-theta) in a right triangle

All the Trig. you need to know for Physics (almost)

Summary

Angle

Opposite (O) side

Adjacent (A) side

Hypotenuse (H)

All the Trig. you need to know for Physics (almost)

SOH, CAH, TOA TrigonometryUsed when working with right triangles only!

Angle

Opposite (O) side

Adjacent (A) side

Hypotenuse (H)

All the Trig. you need to know for Physics (almost)

SOH

sin = O/H

Angle

Opposite (O) side

Hypotenuse (H)

All the Trig. you need to know for Physics (almost)

Example:Find the angle of a right triangle which has a hypotenuse of 12m and a side opposite the angle of 9m.

Angle = ?

Opposite (O) side = 9m

Hypotenuse (H) = 12m

All the Trig. you need to know for Physics (almost)

Example:sin = O/H sin = 9/12 = 0.75 = sin–1 0.75 = 48.6°

Angle = ?

Opposite (O) side = 9m

Hypotenuse (H) = 12m

All the Trig. you need to know for Physics (almost)

CAH

cos = A/H

Angle

Adjacent (A) side

Hypotenuse (H)

All the Trig. you need to know for Physics (almost)

Example:Find the hypotenuse of a right triangle which has an angle of 35° and a side adjacent the angle of 7.5 m.

Angle = 35°Adjacent (A) side = 7.5m

Hypotenuse (H) = ?

All the Trig. you need to know for Physics (almost)

Example:cos = A/H cos 35 = 7.5/H H = 7.5/cos 35° = 7.5/0.819 = 9.16m

Angle = 35°Adjacent (A) side = 7.5m

Hypotenuse (H) = ?

All the Trig. you need to know for Physics (almost)

TOA

tan = O/A

Angle

Opposite (O) side

Adjacent (A) side

All the Trig. you need to know for Physics (almost)

Example:Find the side opposite the 35° angle of a right triangle which has a side adjacent the angle of 7.5 m.

Angle = 35°

Opposite (O) side

Adjacent (A) side = 7.5m

All the Trig. you need to know for Physics (almost)

tan = O/A tan 35 = O/7.5 O = (7.5)(tan 35) = (7.5)(.7)= 5.25m

Angle = 35°

Opposite (O) side

Adjacent (A) side = 7.5m

All the Trig. you need to know for Physics (almost)

SOH, CAH, TOA TrigonometryRemember the Pythagorean TheoremWork on Trig. Worksheet—due tomorrow

Angle

Opposite (O) side

Adjacent (A) side

Hypotenuse (H)

Vector Addition—Algebraic

When adding vectors together algebraically using trigonometry it is important to use the Physics Problem Solving Technique

Vector Addition—Algebraic

Example:A car moves east 45 km turns and travels west 30 km. What are the magnitude and direction of the car’s total displacement?

Vector Addition—Algebraic

Example:A car moves east 45 km turns and travels west 30 km. What are the magnitude and direction of the car’s total displacement?

15 km east

Vector Addition—Algebraic

Example:A car moves east 45 km turns and travels north 30 km. What are the magnitude and direction of the car’s total displacement?

a2 + b2 = c2

c = 54.1 km

magnitude

Vector Addition—Algebraic

Example:A car moves east 45 km turns and travels north 30 km. What are the magnitude and direction of the car’s total displacement?

a2 + b2 = c2

c = 54.1 km

magnitude

Tan = o/a

Tan = 30/45

= Tan-1 0.667

= 33.7

Vector Addition—Algebraic

Example:A car moves east 45 km turns and travels north 30 km. What are the magnitude and direction of the car’s total displacement?

54.1 km

33.7 North of East (?)

Vector Addition—Algebraic

Example:A car is driven east 125 km and then south 65 km. What are the magnitude and direction of the car’s total displacement?Answer:

141 km 27.5 south of east

Vector Addition—Algebraic

Example:A boat is rowed directly across a river at a speed of 2.5 m/s. The river is flowing at a rate of 0.5 m/s. Find the magnitude and direction of the boat’s diagonal motion.Answer

2.55 m/s 11.3 measured from center of river (or 78.7 measured from shore)

Vector Resolution

Vector Resolution—the process of breaking a single vector into its components

Components—the two perpendicular vectors that when added together give a single vectorComponents are along the x-axis and y-axis

Vector Resolution

ExampleA bus travels 23 km on a straight road that is 30 north of east. What are the east and north components of its displacement?

Vector Resolution

ExampleA bus travels 23 km on a straight road that is 30 north of east. What are the east and north components of its displacement?

Vector Resolution

ExampleA bus travels 23 km on a straight road that is 30 north of east. What are the east and north components of its displacement?

cos = a/h cos 30 = a/23

a = 23cos30 = 19.9 km east

Vector Resolution

ExampleA bus travels 23 km on a straight road that is 30 north of east. What are the east and north components of its displacement?

cos = a/h cos 30 = a/23

a = 23cos30 = 19.9 km east

sin = o/h

sin 30 = o/23

o = 23sin30

o = 11.5 km north

Vector Resolution

ExampleA golf ball, hit from the tee, travels 325 m in a direction 25 south of east. What are the east and south components of its displacement?Answer

East (cos 25 = a/325) 295 mSouth (sin 25 = o/325) 137 m

Vector Resolution

ExampleAn airplane flies at 65 m/s at 31 north of west. What are the north and west components of the plane’s velocity?Answer

north (sin 31 = o/65) 33.5 m/swest (cos 31 = a/65) 55.7 m/s

Vector Addition and Resolution

Let’s check our knowledge! (6-10)