hp unit 3 motion in 2d & vectors. consider the following 3 displacement vectors: to add them,...

HP UNIT 3

Motion in 2D & Vectors

Consider the following 3 displacement vectors:

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

d1

d2

d3

Addition of vectors in 2D

d3

d2

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

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

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

Adding vectors using components:

B

A

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

A B

Using vector components instead

A BBy

Bx

Ax

AyWe have already learned how to break down vectors into components

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

Subtraction of Vectors

In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction.

Then we add the negative vector:

Example 1A car drives 26km @ 98° & then 62km @ 30°S of W. Find the resultant displacement of the car. Sketch first.

Example 2

A = 100m, 30.0°      B = 205m, 305° C = 150m, 45° N of W

Determine the resultant displacement (magnitude & direction) of the following vectors A + B - C:

Example 3The eye of a hurricane passes over an island in a direction of 60o north of west with a speed of 41km/h. Three hours later, the hurricane shifts north and its speed slows to 25km/h. Determine the displacement of the eye 4.5h after it passed over the island.

Example 4A pilot wishes to fly due south. A wind is blowing towards the west at 25km/h. The plane can fly at 300km/h relative to the air (in still air). Determine the direction the plane should be pointed to accomplish due south motion.

Example 5: A river flows toward 90.0° relative to shore.  A riverboat captain heads his boat at 297° (starts from western shore side) in order to move directly across the river with velocity 6.0m/s, 0.0° relative to the shore.   

(a) Determine the speed of the river relative to the shore. Draw a head to tail sketch.

Perpendicular Vectors

Perpendicular vectors do NOT affect one another. X does NOT affect Y. Y does not affect X.

A boat can travel @ 3.8m/s relative to river (as if water were still). A boat points itself east straight across a river 240m wide with a river current of 1.6m/s, south. 

(b) Determine the resultant velocity of the boat relative to the shore.   

(a) Determine time it takes boat to cross the river.

(c) Determine how far downstream the boat is by the time it reaches the other side.

(d) At what angle relative to 0o would the boat have to point itself to move straight across the river?

Projectile MotionA projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola.

Horizontal Projectiles (neglect air resistance)

A stone is thrown horizontally at 8.0m/s, 0o from a cliff 78.4 m high. 

Example 1

a) How far from the base of the cliff does the stone strike the ground?

b) Determine the resultant speed of the stone just before it strikes the ground.

A toy car runs off the edge of a table that is 1.225m high.  The car lands 0.400 m from the base of the table. 

Example 2

a) Calculate the speed of the car as it rolled off the table.

b) Determine the vertical speed of the car after 0.25s.

An airplane traveling 1001m above the ocean at 125 km/h is to drop a box of supplies to shipwrecked victims below on a tiny island. Assume no air resistance. 

Example 3

(c) Where will the plane be in relation to the supplies when they strike the island below

(b) What is the horizontal distance between the plane and the victims when the box is dropped?

(a) Where should the plane drop the supplies in relation to the people below?

BEFORE OVER AFTER

Non-Horizontal ProjectilesIf an object is launched at an initial angle of θ with the horizontal, the analysis is similar except that the initial velocity has a vertical component.

An arrow is shot from ground level with initial velocity 40.0 m/s @ 60.0°.  The arrow flies through the air over a level field and then hits the ground.  Neglect air resistance.

a) Determine the range.     

b) Calculate the maximum height.

c) Determine the velocity at 6.0s.

Example 4

A kid kicks a racquet ball off the floor with velocity 8.25m/s @ 76.0°.  The ball hits the ceiling, which is 3.00 m above the floor. 

a) Determine the impact speed with the ceiling.

b) Find the time to reach the ceiling

Example 5

A toy dart gun fires a dart horizontally at the same time another dart is dropped. Both darts start from the same height. Which dart strikes the ground first?

2) In Figure 1, which ball has the greater launch speed?

3) In Figure 2, does ball A and B have the same horizontal speed?

1) In each figure, does ball A spend less, the same, or more time in the air than ball B?

Water leaves the nozzle of a fire hose with velocity 35.0m/s @ 40.0°.  The fire hose is 25.0 m away from the base of the burning building. 

Example 6

b) At what angle does the water stream strike the building?

a) How far up the building does the water hit?

In a game of tennis, you lob a ball with initial speed of 15m/s at 50 degrees. At this instant, your opponent is 10m away from the ball. If the opponent begins moving away from you at 0.3s later, hoping to reach the ball and hit it back at the moment that it is 2.1m above the launch point, with what minimum avg speed must he move to reach it?