Vectors Chapter 4

A quantity with only magnitude Scalar A quantity with only magnitude

A quantity with both magnitude and direction Vector A quantity with both magnitude and direction

Vector Tail Head

The sum of two or more vectors 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

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

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

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

Measure the angle to determine the direction 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 5 - 10 on page 71

Adding Vectors with Opposite Signs Vector1 + (-Vector2) = Vector1 – Vector2

V2 V1 V2 - V1 VR

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 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 R B q A

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 hyp opp q adj

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

R2 = A2 + B2 –2ABcos q otherwise Algebraic R2 = A2 + B2 if right angle R2 = A2 + B2 –2ABcos q otherwise

A ball rolls 45 m north, then is kicked 60. 0 m west 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 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 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 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 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 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

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

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

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

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

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

Determine the x & y components y-axis Determine the x & y components Hypotenuse = 100 m Opposite side = hyp(sin q) q = 37o Adjacent side = hyp(cos q)

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

Resolve the following vector into polar or x & y components: 150 m/s @ 30o N o E

Resolve the following vector into polar or x & y components: 250 N @ 37o E o S

Resolve the following vector into polar or x & y components: 7500 N @ 53o

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

50 m at 45o E o N 2) 45 m at 53o S o W 3) 80 m at 30o W o N 4) 75 m at 37o 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 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 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 37o from horizontal at 1250 m/s Calculate the x & y components.

Check HW: 11 - 14

A jet flies 15 km due west then 25 km at 53. 1o north of west A jet flies 15 km due west then 25 km at 53.1o north of west. Calculate the resultant vector.

Calculate equilibrant 9.0 m W 2) 800.0 cm S 3) 3000.0 mm E 4) 0.0035 km N Calculate equilibrant

Resolve a 2. 4 kN force vector that is 30 Resolve a 2.4 kN force vector that is 30.0o from horizontal into horizontal & vertical components in N:

Calculate equilibrant 2.0 m at 30o 2) 150.0 cm at 37o 3) 3000.0 mm at 53o 4) 0.0040 km at 127o Calculate equilibrant

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

A boat travels at 4. 0 m/s directly across a river flowing at 3. 0 m/s 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 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 53o north of west A jet flies 15 km due west then 25 km at 53o north of west. Calculate the resultant vector.

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

A dog walks 8. 0 m due east then 15 m at 37o north of east A dog walks 8.0 m due east then 15 m at 37o north of east. Calculate the resultant vector.

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

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

A girl walks 25 m due east then 15 m at 37o north of east, the 50 A girl walks 25 m due east then 15 m at 37o north of east, the 50.0 m due south. Calculate the resultant vector.

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

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

Drill: A dog walks: 1) 0.16 km due north 2) 90.0 m due east 3) 25,000 cm at 37o N o E Calculate: Res. & Eq.

Check HW Problems 31 & 31 Page 79

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

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

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

A cat walks: 1) 9.0 m due south 2) 1500 cm due east 3) 5,000 mm at 37o N o E Calculate resultant:

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

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

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

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

Drill & Collect HW: Solve the following: 1) 360 m due west 3) 0.27 km due north Calculate resultant:

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

Use graph paper to solve the following: 1) 3.0 m due west 3) 15 m 53o EoN Calculate resultant:

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

Define the Following: Scalar Vector Magnitude Direction

Define the Following: Distance Displacement Speed Velocity

Test Review

Terms to Define: Equilibrant Vector Resultant Scalar Vector Vector Resolution

Metric Prefixes: Centi Kilo Giga Mega Micro Milli Nano

Trig Functions: Sin q Pytha- Cos q Theorem Tan q Law of Cosines

Add the 3 Vectors Graphically: 50.0 m west 90.0 m north 170 m east

Add the 2 Vectors Mathematically: 20.0 m west 0.10 km @ 37oNoE

Resolve the Vector into x & y comp: 0.450 km @ 53o SoW

Add the 3 Vectors using vector components: 75 m @ 37o NoW 90.0 m @ 37o NoE 150 m @ 53o SoW