Vectors Chapter 4

Scalar A quantity with only magnitude

Vector A quantity with both magnitude and direction

Vector Tail Head

Resultant Vector The sum of two or more vectors

Vector Addition Two addition methods: Graphical Algebraic

Graphical Vector Addition Use the following steps

(1) Draw any one of the vectors with its tail at the starting point or origin

(2) Draw the 2 nd vector with its tail at the head of the first vector

(3) Draw the resultant vector from the starting point of the 1 st vector to the head of the 2 nd

(4) Measure the length of the resultant to determine the magnitude of the vector

(5) Measure the angle to determine the direction of the vector

Drill: An insect crawls 4.0 cm east, then 3.0 cm south. Calculate: a) distance traveled b) displacement

Practice: A plane flies 5.0 km west, then 2500 m south. Calculate: a) distance traveled b) displacement

Drill: A bug crawls 3.0 cm west, then 40.0 mm south. Calculate: a) distance traveled b) displacement

Drill: A plane flies 150 m/s east in a 25 m/s wind blowing towards south. Calculate the plane’s velocity relative to the ground.

Review HW Problems on page 71

Adding Vectors with Opposite Signs Vector 1 + (-Vector 2 ) = Vector 1 – Vector 2

V1V1 V2V2 V 2 - V 1 VRVR

Practice: A bird flies 25 m west, then 57 m east. Calculate: a) distance traveled b) displacement

Practice: A bird flies 14 m west, then 32 m east, then 21 m west. Calculate: a) distance traveled b) displacement

A boat travels upstream at 10.0 m/s in a river flowing at 2.5 m/s. Calculate the velocity of the boat.

Multiple vectors When adding multiple vectors, just repeat the process of head of first to tail of second etc.

Algebraic A B R 

Practice: A car goes 3.0 km west, then 4.0 km south, then 5.0 km north. Calculate: a) distance traveled b) displacement

Algebraic adj opp hyp 

Solving the problem Sin  = opp/hyp Cos  = adj/hyp Tan  = opp/adj

Algebraic R 2 = A 2 + B 2 if right angle R 2 = A 2 + B 2 – 2ABcos  otherwise

A ball rolls 45 m north, then is kicked 60.0 m west. Calculate the distance & displacement of the ball.

A ball thrown at 50.0 m/s north from a train moving 50.0 m/s west. Calculate the velocity of the ball.

A boat travels at 4.0 m/s across in a river flowing at 3.0 m/s. Calculate the velocity of the boat.

A plane travels at 250 m/s south in a 50.0 m/s wind blowing east to west. Calculate the velocity of the plane.

A plane travels at 25 m/s south in a 15 m/s wind blowing east to west. Calculate the velocity of the plane.

Drill: A snail travels at 9.0 cm south then 15.0 cm west then 6.0 cm south. Calculate the displacement of the snail.

Check HW Problems 11 – 14 Page 74

Vector Resolution Resolving any vector into its x & y components

Vector = 100 units at 37 o N o E y-axis x-axis 37 o

Determine the x & y components y-axis Adjacent side 37 o Opposite side Hypotenuse

Solving the problem Sin  = opp/hyp Cos  = adj/hyp Tan  = opp/adj

Solving the problem sin  = opp/hyp opp = hyp x sin 

Solving the problem cos  = adj/hyp adj = hyp x cos 

Determine the x & y components y-axis Adjacent side = hyp(cos  )   Opposite side = hyp(sin  ) Hypotenuse = 100 m

Trig Functions x-component = 100(cos 37 o ) = 100(0.80) = 80 units y-component = 100(sin 37 o ) = 100(0.60) = 60 units

Resolve the following vector into polar or x & y components: o N o E

Resolve the following vector into polar or x & y components: o E o S

Resolve the following vector into polar or x & y components: o

Vector Addition Hint: When adding multiple vectors, just add the vector components. Then solve for the final vector.

1) 50 m at 45 o E o N 2) 45 m at 53 o S o W 3) 80 m at 30 o W o N 4) 75 m at 37 o N o E Calculate resultant

Equilibrium When functions applied to any system add up to zero Steady State Homeostasis

Equilibrant The vector, when added to a set of vectors, would bring the sum of all the vectors back to the zero point or origin.

An automobile is driven 250 km due west, then 150 km due south. Calculate the resultant vector.

A dog walks 4.0 miles east, then 6.0 miles north, then 8.0 miles west. Calculate the resultant vector.

Drill: A cannon fires a projectile at 37 o from horizontal at 1250 m/s Calculate the x & y components.

Check HW:

A jet flies 15 km due west then 25 km at 53.1 o north of west. Calculate the resultant vector.

1) 9.0 m W 2) cm S 3) mm E 4) km N Calculate equilibrant

Resolve a 2.4 kN force vector that is 30.0 o from horizontal into horizontal & vertical components in N:

1) 2.0 m at 30 o 2) cm at 37 o 3) mm at 53 o 4) km at 127 o Calculate equilibrant

The following forces are acting on a point: 1) 5.0 N at 37 o 2) 8.0 N at 53 o Calculate equilibrant

A boat travels at 4.0 m/s directly across a river flowing at 3.0 m/s. Calculate the resultant vector.

A boy walks 4.0 miles east, then 6.0 miles north, then 4.0 miles east. Calculate the resultant vector.

A jet flies 15 km due west then 25 km at 53 o north of west. Calculate the resultant vector.

A jet flies 28 km due west then 21 km north. Calculate the resultant vector.

A human walks 8.0 m due east then 12 m at 30 o north of east. Calculate the resultant vector.

A jet travels 250 miles at 37 o north of west. Resolve the displacement into north & west components.

1) 50 m at 45 o E o N 2) 45 m at 53 o S o W 3) 80 m at 30 o W o N 4) 75 m at 37 o N o E Calculate resultant

A girl walks 25 m due east then 15 m at 37 o north of east, the 50.0 m due south. Calculate the resultant vector.

A girl walks 75 m at 37 o north of east, then 75 m at 53 o west of north. Calculate the resultant vector.

1) 50 m at 45 o S o W 2) 75 m at 53 o E o S 3) 80 m at 37 o N o E 4) 75 m at 33 o W o N Calculate resultant

A zombie walks: 1) 0.16 km due north 2) 90.0 m due east 3) 25,000 cm at 37 o N o E Calculate resultant:

A zombie walks: 1) 0.30 km at 30 o SoW 2) 500 m at 45 o NoE Calculate resultant:

A snail crawls: 1) 25 cm at 37 o WoS 2) 400 mm at 30 o NoE Calculate resultant:

A telephone pole has a wire pulling with a 3500 N force attached at 20 o from the top of the pole. Calculate the force straight down.

A cat walks: 1) 90 m due south 2) 1600 cm due east 3) 5,000 mm at 37 o N o E Calculate resultant:

Forces act on a point: 1) 150 N at 53 o EoS 2) 250 N at 37 o SoW 3) 0.50 kN at 45 o WoS Calculate resultant:

1) 350 N at 53 o WoS 2) 150 N at 37 o NoW 3) 0.25 kN at 45 o WoS 4) 250 N due E Calculate resultant:

1) 0.35 kN due west 2) 150 N due south 3) 0.50 kN at 45 o EoN 4) 250 N at 37 o NoE Calculate resultant:

1) 0.35 kN due west 2) 150 N due south 3) 0.50 kN at 45 o EoN 4) 250 N at 37 o NoE Calculate resultant:

Use graph paper to solve the following: 1) 250  m due east 3) 0.50 mm 53 o EoN Calculate resultant:

Solve with trig: 1) 0.10  N 37 o SoW 2) 250 kN 53 o EoN 3) 150,000 N East Calculate resultant:

Define the Following: Distance Displacement Speed Velocity