332 UNIT 3

Motion in 2D & Vectors

Vector vs Scalar

A VECTOR describes a physical quantity with both ____ and ______.

Examples

A SCALAR describes a physical quantity with _______ only.

Examples

Consider the following 3 displacement vectors:

To add them, place them head to tail where order doesn’t matter

d1

d2d3

Addition of vectors in 2D (not along the same line)

d3

d2

d1 Place vectors head to tail keeping their same orientation (angle).

dR

The sum of d1, d2, d3 is called the resultant, dR.

dR has size and direction. It represents the sum of all the vectors. dR starts at the tail of first

vector and meets head of last vector.

Quadrants and 0o to 360o

Using a rough sketch, estimate the direction of the resultant vector.

Adding vectors using components:

B

A

A and B are generic letters being used to represent these 2 vectors

We have already learned how to add vectors head-to-tail

A B

Using vector components instead

A BBy

Bx

Ax

Ay

Draw components of each vector using head to tail...

A B By

Bx

Ax

Ay

Separate the ‘x’ components

Separate the ‘y’ components

By

BxAxAy

Simply add components as 1D vectors.

Rx

Ry

x’s = Rx and y’s = Ry

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

Ry

Rx

R

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

R Rx2 Ry

2

tan 1 RyRx

Right Triangle Trigonometry

• Sinθ = opposite side / hypontenuse

• Cosθ = adjacent side / hypontenuse

• Tanθ = opposite side / adjacent side

• A2 + B2 = C2Trig functions are just ratios of triangle leg lengths

Resolving vectors into components using trigonometry

Resolve ‘A’ into its x and y components using trig (size and direction)

A = 50m

30o

A football is kicked off a tee at an angle of 40o above the ground. The kicker imparts an initial speed (vi) of 23.0m/s to the ball. Determine the initial speed of the ball in the ‘x’ and ‘y’ directions (vx and vy).

Addition of vectors using component method

Vector A = 30.0m @ 240o

Vector B = 50.0m @ 30o

Vector C = 120.m @ 120o

Example 1: Determine the RESULTANT displacement (size and direction) of A+B+C.

EXAMPLE 2: A roller coaster moves to the right on a level track 50.0m long and then goes up a 25.0m incline of 30o. It then goes down a 15.0m ramp with an incline of 40o. Determine the displacement of the coaster relative to its starting point.

EXAMPLE 3: A boat can move in a river at 15.0m/s using its engine (IN still water). There is a river current of 10.0m/s @ 60o AND a wind blowing due east at 5.0m/s. If the boat points itself at 130o,

b) How far did the boat travel in 1hour?

a) Determine the resultant speed and direction of boat.

Perpendicular Vectors

Plane still travels at 100km/h south even though there is a wind involved to the west.

Motion of vector in x-direction does not alter motion of vector in y-direction.

Example A boat moves with speed 12m/s due west in still water. The current flows at 5.0m/s due north.

a) Draw head to tail vector diagram.

c) If river is 1440m wide, how long will it take to reach other side?

b) Find resultant speed of boat

d) How far downstream will the boat be carried?

EXAMPLE 2A boat rows across a river at 3.8m/s due south in still water. The river flows east at 5.9m/s and is 300m across.

a) In what direction, relative to east, does the boat move?

c) How long does it take to cross the river if there was no current?

b) How far downstream does boat move before reaching shore?

EXAMPLE 3

You are piloting a small plane, and you want to reach an airport 450km due south in 3.5 hours. A wind is blowing from the west at 55 km/h. What heading and airspeed should you choose to reach your destination in time?

Projectile Motion (unit 2 – part2)

Objects that are launched into freefall at an angle (other than 0o or 90o) follow a special path called a parabola.

Projectiles possess BOTH x and y velocities simultaneously

As the ball leaves person’s hand, the velocity of the ball can be broken down into sideways and vertical speeds ( x and y) vix

viy

vi

Initial velocity

vi

θ

Characteristics of Projectiles• Horizontal Direction:

• Vertical Direction:

Horizontal projectiles

Without force of gravity, cannonball would follow horizontal path at constant speed

With gravity added in, ball freefalls downward at ‘g’ with horizontal motion still present where parabola results.

Velocity characteristics for horizontally launched projectile

Projectile Equations

• x – equations:

• y – equations:

example1

A plane flies level at an altitude of 300m with a speed of 100m/s. No air resistance.

B) How fast is bomb moving vertically just before striking ground?

A) If it releases a bomb, how much time would it take to strike the ground below?

• C) Determine the range of the bomb.

• D) Determine resultant speed of bomb just before it hits.

• E) Where is the bomb in relation to the plane at the time it hits ground? (neglect air resistance)

Which way do crates move relative to plane assuming air resistance?

Bullseye?

A tale of 2 bullets

• Consider a horizontally held rifle.• Which hits first, dropped or shot?

example2A rifle held horizontally shoots a bullet at 300m/s at a target. The bullet hits the target 0.12m below the height from which it was fired. How far away was the target from the rifle?

Find the launch speed, v

40m

100m

Angled Projectile characteristics

A person throws a rock along a level surface at 35o with a speed of 25m/s. • A) Find the time for the rock to ascend to it’s peak

in the trajectory.

• B) Determine maximum height for rock.

• C) Calculate the range of the rock.

• D) Find the total hang time (neglect height of release point)

A peach is launched at 60o with a speed of 18m/s.• A) Find the range assuming level surface.

• B) Calculate its resultant speed 3.0s into flight.

• C) Determine height at 2.0s into flight.

A ho-ho is launched at 30o off of a 30m high cliff. The launch speed is 25m/s.

a) Find the total time in the air.

b) Find the resultant velocity right before impact.