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All motion is relative. That is, all quantities of motion are measured relative to a frame of reference.What is considered stationary is dependent on the reference frame of the observer.We can employ vector addition to move our observation from one frame of reference to another.
The measurement fromA to B was produced by a frame of reference with an origin at A.B to C was produced by a frame of reference with an origin at C.
eliminate B as a point of interest, and determine where C is in relation to A.
A boat is travelling such that it is pointing [E] and travelling with a velocity of 4.0 m/s relative to the water.The water is moving over the earth in the northward direction with a velocity of 3.0 m/s.In which direction, and how quickly, is the boat travelling relative to the Earth?(In other words, what is the resulting velocity)
vBW, is the velocity of the boat through the water, 4.0 m/s [E]vWE, is the velocity of the water over the Earth, 3.0 m/s [N], andvBE, is the velocity of the boat relative to the Earth
Find the E-W and N-S components of the following displacement vector: d=530 km [W 25 S]Please determine the magnitude and direction of the total displacement resulting from these two perpendicular motions:d1=43 km [E]d2=22 km [S]An aircraft can travel at a speed of 80 km/h with respect to the air. Determine the resultant velocity of the plane if it encounters a:10 km/h headwind10 km/h tailwind.10 km/h sidewind.60 km/h sidewind.
George V Coast in Antarctica is the windiest place on Earth. Wind speeds there can reach 3.00 x 102 km/h. If a research plane flies into the wind with a speed of 4.50 x 102 km/h relative to the wind, how long does it take the plane to fly between two research station that are 250 km apart?An airplane traveling 325 km/h, at 35˚ North of east encounters a wind blowing 12.5 km/h, South. What is the resultant velocity of the airplane?What distance has the plane traveled in a half-hour?An airplane maintains a heading of due south at an airspeed of 540 km/ h. It is flying through a "jet stream" which is moving at 250 km/h [NE].In what direction is the plane moving with respect to the ground (direction of vPE)?What is the plane's speed with respect to the ground (magnitude of vPE)?What distance does the plane travel in 15 minutes in the direction it is moving relative to the ground?The pilot of a light plane heads due north at an air speed of 4.00 x 102 km/h. A 60. km/h wind is blowing from the west.What is the plane’s velocity with respect to the ground?How far off course would the plane be after 2.5 h, if the pilot had hoped to travel due north but had forgotten to check the wind velocity?A motorboat traveling 5 m/s [E] encounters a current moving at 7.0 m/s, [N].What is the resultant velocity of the motorboat?If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?What distance downstream does the boat reach the opposite shore?A canoeist paddles “north” across a river at 3.0 m/s. (The canoe is always kept pointed at right angles to the river.) The river is flowing east at 4.0 m/s and is 100 m wide.What is the velocity of the canoe relative to the river bank? {Hint: use v }Calculate the time required to cross the river. How far downstream is the landing point form the starting point?A swimmer can swim at a speed of 1.80 m/s in still water. If the current in a river 200 m wide is 1.00 m/s[E], and the swimmer starts on the south bank and swims so that she is always headed directly across the river, determine,the swimmer’s resultant velocity, relative to the river bank. {Hint: use v } how long she will take to reach the far shore. how far downstream she will land (from the point opposite her starting point)?Suppose an airplane flies in a circle of circumference 10.0 km at a constant speed of 100 km/h ± 0.5 km/h.What is the change in velocity of the plane in one quarter of a revolution?What is the change in velocity of the plane in one half of a revolution?
An airplane maintains a heading of due south at an airspeed of 540 km/ h. It is flying through a "jet stream" which is moving at 250 km/h [NE].In what direction is the plane moving with respect to the ground (direction of vPE)?What is the plane's speed with respect to the ground (magnitude of vPE)?What distance does the plane travel in 15 minutes in the direction it is moving relative to the ground?A boat can move at 9.0 km/h relative to the water. The boat's pilot keeps the boat pointed at right angles to the river bank. The river is flowing at 6.0 km/h and is 0.20 km across.In what direction does his boat actually go relative to the shore?How long does it take him to cross the river?How far is his landing poInt downstream from his starting point?How long would it take him to cross the river if there were no current?To go straight across the river, at what angle should the pilot point the boat?An airplane is flying toward a destination 2.00 x 102 km due east of its starting point (desired displacement vPE), and the wind is from the northwest at 30.0 km/h vWE. The pilot wishes to make the trip in 40.0 minutes [time interval].What should the plane's heading be?At what airspeed should the plane fly?A swimmer on the south shore of a river wishes to swim to a dock due north of his starting point. His maximum swimming speed in still water is 4.0 km/h, and there is a current in the river flowing at 2.5 km/h towards the west.In what direction must he set out and continue swimming through the water? If the river is 2.0 km wide, how long does it take him to make the crossing?A helicopter flying where the average wind velocity is 38 km/h [25° N of E] or [E 25° N], needs to achieve a velocity of 91 km/h [17° W of N] or [N 17° W] relative to the ground to arrive at its destination on time. What is the necessary velocity relative to the air?The navigator of an airplane plans a flight from one airport to another 1200. km away, in a direction 30° East of North. The weather office informs him of a prevailing wind from the west at 80. km/h. The pilot wants to maintain an airspeed of 300. km/h.What heading should the navigator give the pilot?How long will the flight take?
A novice paddler would like to travel due East across a river, however the river is flowing with current 2.7 m/s [E 25º S]. If their maximum paddling speed is 2.0 m∕s…Is it possible for the resulting velocity vbe to solely point in the Eastward direction?If they can and they must travel 1.5 km [E], how long will it take?In order for an airplane to arrive at its destination on time it must fly with a velocity of vpw = 425 m/s on a heading of [W 25.0º N]. On this particular day the wind is blowing South with a velocity of vwg = 135 m/s.What will be the resulting magnitude of the velocity of this airplane?What will be the resulting direction of the velocity of this airplane? What will be the displacement [give both magnitude and direction] of this plane after flying with this velocity for 2.5 hours? CHALLENGE: What should the velocity [give both magnitude and direction] of this airplane be if it is to arrive at its destination on time?A plane is traveling with an airspeed of 500.0 km/h. A wind is blowing toward [N 30˚ E] at 60.0 km/h. If Hamilton is [S 50˚ E] of the plane, then in what direction should the plane head in order to fly a straight course to Hamilton? What is the velocity of the plane relative to the ground? (must use sine/cosine law)A girl riding a bicycle at 4.0 m/s [S 15˚ W] is delivering newspapers. She threw a paper with a horizontal velocity of 11.0 m/s [W 35˚ N] relative to the ground. The paper was in the air for 0.644 s. Assume that there is no friction or wind.Please calculate the horizontal velocity of the newspaper relative to the bike.Please calculate the paper's initial vertical velocity.Please calculate the paper's initial velocity, relative to the ground, in the vertical plane.Robin Hood is traveling by train through Sherwood Forest on his way to Nottingham, with a constant velocity of 20.0 m/s [S] with respect to the ground. As he passes a crossroad he spies an evil knight resting beside his trusty Kawasaki at 300.0 m [E] from the rail line. After taking 5.00 seconds to load his bow, Robin shoots it from the train with a horizontal speed of 50.0 m/s (we can ignore the fact that he shot the arrow at 45.0° above the horizontal) Determine the velocity (horizontally) of the arrow along the path from Robin to the knight. and the direction in which Robin has to aim to take into account the train's motion. (must use sine/cosine law)
Relative Motion
Applying Vector Components
Relative Motion
Example:
You start at position A, and move to position B by travelling 4.0 km [E] and then move to position C by travelling 3.0 km [N].
What is the displacement from A to C?
We looked at this in the last lesson, and found that the overall displacement can be found by using the Pythagorean Theorem and our Primary Trigonometric Ratios, and discovered that the resulting displacement is d=5.0 km [E 37 N]
Why does this work?
And, algebraically, this would look like:
dAC=dAB+dBC
By adding the vectors in this manner, we:
This doesn’t just apply to displacement vectors…
We can apply this to velocities as well.
Algebraically, this would look like:
vBE=vBW+vWE
Where:
We would go through the process of adding the magnitudes with Pythagorean Theorem, and determining the direction with , to produce:
vBE=5.0 m/s [E 37 N]
And because of the skills we learned earlier with vector components we can solve problems with any number of non-perpendicular vectors.
Tips:
Organize your solutions in a table.
Vector
x-component
y-component
vBW
4.0 m/s
0
vWE
0
3.0 m/s
vBE
4.0 m/s
3.0 m/s
There are no rows in this table
Worked Example
Completion Problem
Navigation Problems
All of these problems involve the addition of vectors and constant velocity kinematics. Please solve them using components of vectors unless otherwise noted.
Level 1
Level 2
Level 3
Level 4
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