Vectors – Day #1
A vector is a quantity with both _____________ (a size or number) and a
_____________ (which way it is pointing). A scalar is a quantity with ___________ only.
For example, Velocity is a vector v = 40 m/s [North 30o EAST].
Speed is a scalar
s = 40 m/s.
How are vector “directions” given?
40 m/s
30
10 m
60o
o
20 m/s
10o
Vectors
Scalars
When vectors are drawn “graphically”, notice that
the length of the arrow used is _____________ to
the magnitude of the vector (40 m/s is twice as long
as 20 m/s). The table at the right lists some common
vectors and scalars.
Vectors can be added or subtracted. When they are, the result is another vector, known
as the __________________ vector. We always add vectors __________________.
5 km
3 km
=
+
10 m/s
then
+
5 m/s
Consider a person walking 40 km North and
60 km East.
=
40 km
60 km
To determine the displacement of the walker, add the two vectors ____________. The
“resultant” will be from the start to the finish of the added vectors. This is known as the
resultant vector, “R.” I usually EMBOLDEN this vector so that it stands out.
60 km
60 km
40 km
Vectors to be
added together
40 km
Resultant
vector
The distance covered by the walker is _______. Note that the distance is a ________
and does not get a direction, only a magnitude. The displacement is another story. Using
the ___________________, we can determine the length of the displacement
(resultant). It will come out to _____________________. This is the magnitude of the
displacement vector. Now we need to find the direction of the new vector. We will
always describe the angle in which the vector is pointing in the ____________corner of
the drawing. This is where the tail of the resultant meets a tail of one of the original
vectors. This angle gives the direction of the new “Resultant” vector.
We use _______ alphabet symbols to
represent angles as variables and regular
lowercase letters such as x, y, z, etc to
represent numbers as variables. To find we
must use trigonometry ( ____ ____ ____ )
Find the value of theta in the space below.

If you got .9827 for your answer, your calculator is in ___________ mode. Be careful
of this. We will report this answer for displacement as follows:
___________________________
This has now, both magnitude and direction. It is a complete vector. The walker was
displaced this much and in this direction.
Vectors may also be used to describe a summation of velocities. Consider if you jumped
into a river with a current flowing to the right at 2 m/s and you swam pointing directly to
the other shore, moving through the water at 6 m/s. Yes you will be blown downstream.
There are two vectors involved. Your water speed 6m/s, and the current speed 2m/s.
Current speed
(2 m/s)
Water speed
(6 m/s)
Resulting
velocity
6 m/s
Notice how the
vectors were
arranged
__________.


2 m/s
You will move North East as shown here with respect to the shore. The angle in here will
be ______________ and the hypotenuse will be __________________________.
This value is known as the ___________________. Every vector can be broken down
into its ____________, which are in the x & y directions. In the previous examples, the
vectors used were relatively simple vectors since they were pointing in purely N,S,E, or
West directions. These vectors still had both x & y components, but either their x
component or their y component were _______.
Example:
Current speed
x component: 2 m/s
y component: 0 m/s
Water speed
x component: 0 m/s
y component: 6 m/s
However, the answer to the “swimming” problem (the resultant vector) was “crooked”. It
had both an x & a y component.
HW #1 Vectors Problems
Graphical Vectors
1. Add the two vectors at the
right graphically
50 m/s
20 m/s
2. A stray dog wanders 4 km west, 2 km North, 3 km west, and finally 3 km south. Draw his journey using vectors and then
draw his displacement vector and find its magnitude.
3. A plane flies 9 km north, then flies 6 km S 30 o E. Draw the plane’s displacement vector from the start of the trip.
4. Use the vectors below to “Graphically” show each of the following.
A
a) A + E
B
b) D + B
C
c) 3C
D
E
d) B + 2D
1D Vectors
5. A man walks 9 km east and then 15 km west. Find his
a) distance traveled
b) displacement
2D Vectors
6. A robin is flying south for the winter at a rate of 40 miles per hour when it runs into a hurricane blowing due west at 100
miles per hour. What is the new, “resulting” velocity of the robin?
7. Tarzan is hanging from one of his vines. His weight (his force) is 210 pounds, directed Southwards. A boy sees that
Tarzan wants to be pushed on his vine. The boy pushes with a force of 90 pounds West. The rope can withstand a 225lb force. What happens to the “King of the Apes”? (Hint: First find both the magnitude and the direction of the
“resultant”).
8. On a television sports review show, a film clip is shown of the great tackle that ended the 1960 championship football
game in Philadelphia. Jim Taylor of Green Bay was running forward down the field for a touchdown with a force of
1000 newtons (direction = South). Check Bednardek of Philadelphia runs across the field “at right angles to Taylor’s
path” (direction = West) and tackles him with a force of 2500 newtons. Find the resultant of this crash.
9. A kite weighs 5 newtons (downward force). A girl throws the kite straight UP in the air with a force of 20 newtons. If the
wind is blowing horizontally with a force of 20 newtons EAST, find the resultant force acting upon the kite. Find both
its magnitude and its direction.
Vectors – Day #2
When finding displacement or average velocity in 2D, we need to
remember the equations
___________
____________
____________
Example:
a) A certain man’s initial position is 40 m [N], as measured from his house. After 30
seconds of walking, his new position is 50 m [S]. Find:
i) the distance travelled by the object during the trip.
ii) the man’s displacement.
iii) the man’s average velocity.
b) A radar station is tracking a jet. Its location at a certain moment on time is
40 km [W] at an altitude of 5000 m, relative to the station. 2 minutes later, its
location is 53 km [E], this time at an altitude of 3000 m. Find the jet’s
i) displacement.
ii) average velocity.
Resolving a vector into components
Every vector can be written as a combination of only x-direction and y-direction vectors. These vectors are
called the __________________ of the original vector. When a vector is broken into its components, we
call this action “_____________ the vector into components”.
For each vector below, 1st name the vector. Then resolve the vector into its components.
a)
c)
60 km
b)
50o
50 m/s
20o
3N
d) A bird is flying at a speed of 10 m/s at an angle of 30 o below the horizontal. At what speed is
the bird’s shadow moving relative to the ground.
e) A small airplane takes off at a constant velocity of 150 km/h at an angle of 37 o above the
ground. How high above the ground is plane after 3 seconds of flight? What horizontal
distance does it undergo during this time?
#2 Vectors HW Problems
2D Vector Problem
10. The world is again being attacked by hostile aliens from outer space. This looks like a job for
Superman! With his super strength, he is able to fend off the attack. However, he wants to rid the
earth of these dangerous aliens forever. As they fly away, they must pass close to our sun. They are
heading on a bearing of East with a speed of Worp 8. If Superman can change their course to a
bearing of between
[N 61o E] and [N 64 o E], they will crash into the sun. He gives them a
velocity of Worp 4 on a bearing North. What happens? Has Good triumphed again? (Hint: Find
the resultant worp speed and see if its direction is in the appropriate range)
Resolving Vectors into Components
45 km
11. Resolve the vector at the right into its vertical and
horizontal components.
40o
12. Resolve the following vectors into their components.
a) 55 m [N 40o W]
b) 0.4 km/h at 60o W of S
c) 19N [SE]
13. Wonderwoman, in a feat of super strength, pulls a sled loaded with 20 kids over the sand at the beach
(she’s a little early for the winter but can’t wait) with a force of 2,000 lbs. The rope makes a 20 o
angle with the horizontal. Find the components of the rope.
14. A VW is parked on a hill with a slope of 30o. The car has a downward force of 9,000
Newtons acting on it (otherwise known as its weight). Find the component
forces. (Hint: One component is parallel to the slope and the other is
30o
perpendicular to the slope).
15. A man pushes a lawn mower with a 500-newton force. The handle makes a 40o angle with the
ground. Find the horizontal and vertical components of his applied force.
16. Another man pulls a wagon with his son sitting in it. The man pulls the wagon with a force of 200
N at an angle of 40o with the ground. The sidewalk provides 30 N of friction in the opposite
direction of the man’s motion. What is the net force on the wagon in the direction parallel to the
ground?
Avg Velocity Problems
17. A baseball is hit from homeplate into the outfield bleachers. It lands 135 meters from the plate,
horizontally, and 10 meters vertically above the field. If the ball is in the air for 5 seconds, find:
a) the average velocity of the ball during the trip (Both magnitude and direction).
b) the average velocity of the ball’s shadow upon the field during the trip (magnitude only).
Vectors – Day #3
When vectors are “crooked”, they become slightly more complicated to add together. For
example, consider a car driving 20 km [West] and then 50 km [West 30 o South]. They
are still added head to tail. Vectors add ________________. You could add the 20 to
the 50 and get the same resultant as if you added the 50 to the 20. An efficient
practice is to draw any pure N,S, E, or W vector first and then add the “crooked” vector.
We have to break the vectors down into components. The 20 km is already fine. The 50
km must be broken down before we try to use the Pythagorean Theorem.
Method I
Component Method
30o
+
50
20
The 50 km vector is now represented in pure x and y values. The 20 km vector is still
fine. Now we have to add up all of our x values (+ right, - left) and all of the y values (+
up, - down).
x:
(the zero is to remind you that the
first vector had no donation to the y)
y:
63.3
Now, recombine the x and y:
25
Note that the –63.3 means the vector should point left and the –25 means the vector
should point down. Next, the resultant may be drawn in.
63.3

25
d =
68.1 km

or
To give the proper direction (South 21.6o West or West 21.6o South) you must follow the
component of the vector whose direction you are giving. Notice that we could have added
the components in either order, thereby getting two different answers. The angle
included in the directions is always the angle included between the “starting” component
and the resultant.
Practice: Name each resultant vector below in two (2) different ways.
46.2o
7 m/s
12 ft
20o
Method II
Graphical Method
Go back to the example where we started with the resultant drawing of
20 km
20 km
300
+
50 km
50 km
=
We will now enclose the drawing to the right in a rectangle.
50 cos 30
20
300

50 sin 30
Solving: x = ________________________

y
R
y = ________________________
x
Calculating the magnitude of the resultant, we get _________________.
We have a choice for describing the angle of direction, either  or 

=
=
We could report our answer one of two ways:
____________________________
_______________________ or
In-Class Examples
When adding crooked vectors, we use the following procedure:
1)
2)
3)
4)
5)
6)
Draw all the vectors appropriately.
Resolve each vector into its components.
Add the x-components together (making sure to keep track of signs), yielding SUPER-X.
Add the y-components together (making sure to keep track of signs), yielding SUPER-Y.
Use SUPER-X and SUPER-Y to construct a SUPER-TRIANGLE.
Find the resultant (magnitude and direction) of the SUPER-TRIANGLE.
a) A box is pulled with forces of 45 N [W], 100 N [W 40o N] and 200 N [E 20 o S]. Find the net force
on the box.
b) A stationary quarterback is hit simultaneously by 3 defensive players. They hit him with forces of
300 lb [W 20o S], 400 lb at 50 o North of East, and 200 lb [N]. Which direction will the
quarterback move after this collision?
c) Three vectors are acting on a box, yielding a resultant force of 80 Newtons [N]. Two of the
vectors acting on the box are 5 Newtons [S] and 42 Newtons [SW]. Find the 3rd force.
#3 Vectors HW Problems
18. Two ants at a picnic find a small piece of cake. Each picks up the cake and tries to carry it home.
The first ant’s home is at a bearing of [W 15o N] and he pulls with a force of 10 dynes. The second
ant’s home is at a bearing of [S 15o W] , and he pulls with a force of 13.5 dynes. In which direction
will the cake travel?
19. Linda is pulling a sled with a force of 20 pounds and is heading on a bearing of N30 E. Elaine
begins to pull the sled with a force of 15 pounds on a bearing of N5 W. What net force is exerted
on the sled and in which direction does it go?
20. Find the resultant of a 120-Newton force North and an 80 Newton force at N45 E.
21. Two water skiers are being pulled by a boat. The first skier exerts a force of 300 pounds North on
the boat; the second skier exerts a force of 400 pounds at N75 E. Find the resultant of the two
forces.
22. Three brothers are playing at the local playground with a frisbee. Each grabs it at the same time.
Robbie grabs it at the N63 E point and exerts a force of 100 pounds on it. Chip grabs it at the N27
W point and exerts a force of 100 pounds on it. Ernie pulls with 141.4 pounds at the S18 W point.
Who gets control of the frisbee?
23. Back to problem #18. If a third ant started pulling on the piece of cake, what would his force have
to be in order to keep the cake from moving? In other words, what 3rd force would make both the x
and y-components sum to zero.
Vectors – Day #4
Equilibrium Problems are problems in which an object _________________________.
In these problems of values of the x-components of all the vectors involve sum to
_______. This is also true for the sum of the values of the y-components.
Examples:
a) Four forces pull on an object, and the object doesn’t move. The first three forces are 40 N [E],
65 N [W 50o S], and 80 N [N 10o W]. Find the magnitude and direction of the 4 th force.
b) The “Great Houdini”, in one of his death-defying acts, was suspended from a flagpole while tied in a
strait jacket. The flagpole is supported by a cable which makes a 60 o angle with the pole. What is
the actual weight that the pole is supporting if the force on the cable is 750 lbs. (Maybe Harry
should go on a diet!!!!)
BUILDING
cable
60o
flagpole
c) A traffic light is supported by two cables that are 140 apart. Each cable exerts a force of 150
Newtons on the light. How much does the light weigh? (Think of the weight of the light, the
downward force, as the equilibrant.)
#4 Vectors HW Problems
24. Four boys are playing tug-o-war. Three are pulling with forces of 7 Newtons [N], 17 Newtons [S],
and 41 Newtons [S]. If the rope is not moving, find the force of the fourth boy.
25. Three forces act on an object, which is equilibrium. The first two forces are 60 lbs [NW] and
33 lbs [E]. Find the 3rd force.
26. A speaker in the auditorium is supported by two cables that are 120 apart. If the speaker weighs
200 Newtons, find the tension in each of the cables. (Hint: Look at example problem “c” above to
get a general problem solving strategy.)
27. What is true of the sum of the x-components of the vectors that act on an object that is in
equilibrium?
28. Twenty forces act on an object that is in equilibrium. If the vector sum of the first nineteen forces is
4500 N [E 20 N], find the magnitude and direction of the twentieth force.
29. Three vectors are added together and the resultant is 50 km [W]. The first two forces are 30 km [E]
and 65 km [SW]. Find the magnitude and direction of the 3rd force.
Vectors – Day #5
When finding the change in velocity or average acceleration in 2D, we need to remember
the equations
____________
____________
Examples:
a) A ball is thrown at a wall at a speed of 30 mph. It loses some of its energy in the collision, and
bounces off the wall at a speed of 25 mph. If the ball deformed during the collision and was in
contact with the wall for 30 ms, find the average acceleration of the ball during the collision.
b) A hockey puck hits the boards with a velocity of 10 m/s E20 S. It is deflected with a velocity of
8.0 m/s at E24 N. If the time of impact is 0.03 s, what is the average acceleration of the puck?
8 m/s
10 m/s
20
24
#5 Vectors HW Problems
30. A man was travelling West at 4 m/s is now travelling East at 9 m/s. Find his change in velocity
during this timer period.
31. Do problem #30 again, this time assuming that the man’s final velocity is 9 m/s North.
32. An object travels west @ 90 m/s. It changes direction (in 5 seconds) and then travels @ 100 m/s
[N 30o E]. Find the average acceleration of the turn.
33. A boy travels @ 10 m/s [S] for 1 minute and then @ 5 m/s [W 10 o N] for 2 minutes. Find his
average velocity for the trip.
34. A man travels 10 km [N], then 50 km [E 30 o S], then 100 km [W], and finally 30 km [W 40 o S].
If the trip takes 4hrs, find the:
a) distance traveled.
b) displacement for the entire trip
c) difference between the magnitudes of s and v .
35. An object once moving at a velocity of 15 m/s [S] is now moving at a velocity of 25 m/s [E], 15
seconds later. Find the object’s change in velocity during the period of time as well as its
average acceleration.
Vectors – Day #7
Relative Velocity
Example: A man is standing on a riverboat next to his wife. The boat is moving down
a river without propelling itself, using only the current’s speed, which is
50 ft/min. The man starts jogging towards the front of the boat with a
speed of 100 ft/min. During his jog, a smaller speedboat, traveling at
150 ft/min (relative to the water), passes the boat (in the same
direction). A bird is sitting on the shore, watching the whole situation
unravel.
•
What is the man’s velocity (relative to the bird) when he is running?
_____
•
What is the man’s velocity (relative to his wife) when he is running? _____
•
What is the man’s velocity (relative to the speedboat) when he is running? _____
•
What is the wife’s velocity relative to the bird? _____
•
What is the wife’s velocity relative to the speedboat? _____
Example: Two cars are moving towards each other on a highway. Car A moves East at 60 mph, while car B
moves West at 50 mph. Find the velocity of car A with respect to ….
a) car B ______
b) a bird sitting on the side of the highway ______
c) a boy in the backseat of car A ______
The Relative Velocity Equation
river boat relative to bird
vMB  vMR  vRB
vPA  vPG  vGA
man relative to bird
man relative to riverboat
Package relative to airplane
Example:
Ground relative to airplane
Package relative to ground
A boy sits in his car with a tennis ball. The car is moving at a speed of 20 mph. If he can throw the ball 30
mph, find the speed with which he will could hit…
a) his brother in the front seat of the car.
b) A sign on the side of the road that the car is about to pass.
c) A sign on the side of the road that the car has already passed.
River Problems
A 20 m wide river flows at 1.5 m/s. A boy canoes across it at 2 m/s relative to the water.
a) What is the least time he requires to
cross the river?
b) How far downstream will he be when
she lands on the opposite shore
(assuming he tries to cross in the
least amount of time)?
c) What will his velocity relative to the
shore be as he crosses?
A river is 20 m wide. It flows at 1.5 m/s. If a girl swims at a speed of 2 m/s, find:
a) the time required for the girl to swim 15 m
upstream (assuming she points the canoe
directly upstream).
b) the time required for the girl to swim 20 m
downstream (assuming she points the canoe
directly downstream).
c) the angle (between the swimmers path and
the shore that the girl should aim when
crossing the river if she wants to arrive at
the other side directly across from her
starting point.
d) How long will it take to cross the river in
this case (the case where the girl adjusts
for the current by pointing herself
upstream)
#7 & #8 Relative motion HW Problems
Level I
36. Car A is driving at a speed of 30 mph to the right. Car B is driving at a speed of 40 mph to the left. The
car’s are moving toward each other. What is the velocity of ….
a)
b)
c)
d)
car B with respect to car A?
car A with respect to car B?
car A with respect to a police officer parked on the side of the road?
Car A with respect to someone in the passenger seat of car A?
37. A boy is sitting on an airplane (made entirely of see-thorough plastic) that is travelling west at 300 mph.
He throws a ball toward the back of the plane. If he throws the ball at 30 mph, what is the velocity of the
ball relative to …
a) a boy sitting in back of him on the plane?
b) a boy sitting on the ground watching the event from a lawn-chair?
38. A 30 m wide river flows at 1 m/s. A girl swims across it at 2 m/s relative to the water.
a) What is the least time she requires to cross the river?
b) How far downstream will she be when she lands on the opposite shore?
c) If the girl puts her head down and tries to swim straight across the river, at what angle
relative to the shore will she actually travel? What will be her speed relative to the shore?
39. A plane is flying through the air at a speed of 400 mph [E] when it encounters a 50 mph wind that is
pointing North. Find the plane’s velocity with respect to the ground while it fights the wind.
40. Delivering a paper, a paper boy is riding his bicycle with a velocity of 12 m/s [E]. If he throws the paper
with a velocity of 20 m/s [N], what will be the velocity of the paper with respect to the ground?
41. A field goal kicker attempts a field goal. He kicks the ball with a velocity of 25 m/s directly North
toward the goal post. However, there is a wind blowing in the stadium from east to west with a velocity
of 6 m/s. What will be the velocity of the ball with respect to the ground?
Level II
42. A 70 m wide river flows at 0.80 m/s. A canoeist (who can paddle the canoe at 2.4 m/s in still water) sets
out from shore. At what angle to the shore would the canoe have to aim, in order to arrive at a point
directly opposite the starting point? How long would this trip take?
43. An airplane maintains a heading due West at an air speed of 900 km/hr. It is flying through a hurricane
with winds of 300 km/hr [S45 W].
a) Find the plane’s velocity (magnitude and direction) relative to the ground.
b) How long would it take the plane to fly 500 km along the path in part “a”?
44. A river is 50 m wide. It flows at 2 m/s. If a canoeist can paddle at a speed of 5 m/s, find:
a) the time required for the canoeist to paddle 20 m upstream.
b) the time required for the canoeist to travel 100 m downstream.
c) the angle (between his canoe tip and the shore) that a canoeist should aim when crossing the river
if he wants to arrive at the other side directly across from his starting position.
45. A plane has an airspeed of 100 m/sec. The pilot notices that although he was headed due East, a wind of
80 m/s North is pushing the plane. What is the plane’s velocity relative to the ground?
46. 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.
a)
b)
c)
d)
What is the velocity of the canoe relative to the river bank?
Write the relative velocity equation that was used to solve part “a”.
Calculate the time required to cross the river.
How far downstream is the landing point from the starting point?
47. A boat which can travel at 5 m/s in still water attempts to cross a river by aiming straight across. It
takes the boat 20 seconds to cross the river and the boat lands 10 meters downstream. Find….
a)
b)
c)
d)
the width of the river.
the speed of the current.
the speed that the boat’s speedometer will read during the trip.
the angle relative to the shore that the boat should point itself so that it actually travels straight
across.
e) the time the trip in part “d” will take to cross the river.
Level III
48. While standing in the pocket in Sunday’s big game, Donovan McNabb throws a pass downfield. He
can throw the ball with a velocity of 22 m/s. If he wants the ball to go directly South to the end
zone, which way (a specific direction) should he throw the ball if there is a wind blowing from east
to west with a velocity of 8 m/s?
49. A kayak can move at a speed of 5 m/s in still water. How long will it take the kayak (roundtrip) to travel
100 meters upstream and then back to its starting position if the speed of the current is 3 m/s?
50. A plane is flying with an airspeed of 200 mph [N]. It encounters wind that changes its course, causing it
to travel at a speed of 230 mph in a direction [N10oW]. Find the magnitude and direction of this wind’s
velocity.
Days #9 & #10
51.
What is a vector. Fully explain in words.
52.
Find the resultant vector of each set of vectors below.
a.
b.
c.
53.
45 lb [E] & 70 lb [ N]
105 m/s [S] & 200 m/s [E 40o N]
80 km [N 20o W] & 350 km at 70 o south of west
If a man walked 80 m South in 2 min and then 40 m West in 1.5 minutes, find his:
a.
average speed
b) average velocity
54.
Find the change in velocity vector for an object that was traveling at 50 mph due east and is now
traveling 30 mph due west.
55.
Find the average acceleration of an object that took 5 seconds to change its velocity from 60 m/s
[S] to 70 m/s [E 20o N].
56.
If two people pull an object with forces of 200 Newtons [S 50o E] and 150 Newtons [N 10o W], what
force would a 3rd person have to pull in the direction [S 40o E] in order to keep the
object from moving North or South? (It can move East or West, but not North or
South).
57.
Describe in words what must be true for an object to be in equilibrium. Refer to the forces that are
acting on the object.
58.
A man pushes a box with a force of 100 lb at an angle of 40 o above the horizontal. Another man, on
the other side of the box, pushes back on the the box with an unknown force at angle of
20o above the horizontal. The box doesn’t move. What is the unknown force?
59.
Adam takes Carly’s hat and throws it 10 m [West] to Amanda who throws it to 5 meters [South] to
Rachel, who finally throws it 6 meters [SouthEast] to her teacher. Determine the
displacement of the hat.
60.
A passenger wishes to throw a soda bottle at 8 m/s into a trash can located 24 meters from the side
of the road while continuing to drive at 4 m/s. If the passenger releases the bottle
perpendicular to the path of the car, how far (distance) in advance should she release
the bottle?
The ULTIMATE river problem
A river is 100m wide. The current is flowing at 3 m/s. Canoeist Bob paddles at 5 m/s, canoeist
Joe paddles at 4 m/s, and canoeist Sally paddles at 6 m/s (all relative to the water).
a)
b)
c)
d)
If Bob tries to cross the river perpendicularly, how long will it take him?
How far downstream will Bob land after crossing the river?
What is Bob’s velocity relative to the shore?
Write the relative velocity equation.
e) If Joe wanted to cross the river and land at a point directly across from his starting point,
what direction (relative to the shore) would he aim his canoe?
f) How long would it take Joe to cross the river (and land at a point directly across from
him)?
g)
h)
i)
j)
How long will it take Sally to canoe 50 m upstream?
What is Sally’s velocity relative to a bird sitting on the shore?
If Bob paddles for 50 seconds downstream, how far will he travel (relative to the shore)?
If Sally (paddling upstream) passes Bob (paddling downstream), what is Sally’s velocity
relative to Bob?
k) What is Bob’s velocity relative to Sally?
l) If Joe and Sally set out together on a trip downstream, what would Sally’s velocity be
relative to Joe?
m) What would Joe’s velocity be relative to Sally?
n) If Joe wanted to canoe 200m upstream and then back to his starting point, how long would
it take him?
o) CHALLENGE (don’t get discouraged….its not easy….and its much too hard for the test): If
Sally wanted to cross the river (in a straight line) and land at a point 20 m downstream,
what is the angle that she should point herself relative to the shore?
Ultimate Plane Problem Answers (on next page)
a)
802 km/h [N 3.6o E]
c) 1603 km
e)
[N 3.6o W]
g) 3.792h (3 hrs, 48 min)
b)
v PG  v PW  vW G
d) 100 km too far east
f) 264 km/h [W 47o S]
h) 1,019 km/h [E 77o N]
The ULTIMATE plane problem
On a certain night, the wind is blowing from the west at 50 km/h. A Boeing 747 flies with an
airspeed of 800 km/h due North while a small private plane flies at 300 km/h at a heading of
[W 40o S].
a)
b)
c)
d)
e)
Find the velocity of the Boeing 747 with respect to the ground.
Write the relative velocity equation for this situation.
What distance will the plane cover if it flies for 2 hours?
If it wanted to head due North, how far off course will it be after these 2 hours?
What angle should the pilot redirect the plane in order for the plane to head due North?
~~~~~~
f) Find the velocity of the private plane with respect to the ground.
g) How long will it take for the plane to fly 1000 km (relative to the ground) in a straight line?
~~~~~~
h) Challenge (again, don’t get discouraged….its not easy): Find the velocity of the Boeing 747
relative to the private plane.
(Hint: Start off by writing a relative velocity equation. Then, use your answers from parts
“a” and “f” above)
Ultimate River Problem Answers (on previous page)
a) 20 sec
b) 60 m
c) 5.8 m/s [across 31o downstream]
d) vCB  vCW  vW B
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
[across 49o upstream] or [upstream 41o across]
37.8 sec
16.7 sec
3 m/s upstream
400 sec
11 m/s upstream
11 m/s downstream
2 m/s downstream
2 m/s upstream
229 sec
[across 18o upstream] or [upstream 72o across]
Answers to this packet
2)
7.07 km
5)
a: 24 km
b: 6 km [west]
6)
108 mi/hr @ [S 68 W] or [W 22 S]
7)
228 lb. @ [S 23 W] or [W 67 S]
34) 190 km, 86.74 km [W 23.3 o S]
Savg = 47.5 km/h, Vavg = 21.7 km/h,
diff = 25.8 km/h
8)
2693 N @ [W 22 S] or [S 68 W]
35) 29.155 m/s [E 31o N], 1.94 m/s2 [E 31o N]
9)
25 N @ 53 from vertical or 37 from horizontal
10)
8.94 worps @ [N 63 E] or [E 27 N]
11)
x: 34.47 km;
12)
-35.353 m, 42.132 m;
13.44 N, -13.44N
32) 32.92 m/s2 at 31.7o N of E
33) 4.285 m/s @ 40o S of W
y: 28.93 km
13)
Horizontal = 1,879 lbs
Vertical = 684 lbs
14)
Parallel = 4,500 N
Perpendicular = 7,794 N
-.35 km/h, -0.2 km/h;
36) 70 mph [L]; 70 mph [R]; 30 mph [R]; 0 mph
37) 270 mph [forward]
38) 15 sec; 15 m; 2.24 m/s; DS 63.4 o Across
39) 403.1 mph [E 7.13o N]
40) 23.3 m/s [E 59 o N]
41) 25.7 m/s [N 13.5 o W]
42) U 70.53 o A; 30.97 sec
43) 1132.2 km/h [S 79 o W]; 0.44 h
15)
321.4 N [downward], 383 N [in motion direction]
44) 6.67 sec; 14.29 sec; U 66.4 o A
16)
123.2 N [in motion direction]
45) 128.06 m/s [E 38.7 o N]
17)
27.1 m/s [ 4.2o up], 27 m/s
46) 5 m/s [A 53.1 o DS]; 33.3 sec; 133.3 m
18)
[W 39o S]
47) 0.5 m/s; 100m; 5.03 m/s [A 5.7 o DS];
[U 84.3 o A]; 20.1 sec
19) 33.4 lbs. @ [N 15 E]
48) [S 21.3 o E]
20) 185 N @ [N 18 E]
49) 62.5 sec
21) 559 lbs. @ [N 44 E]
22) Nobody get the frisbee. It doesn’t even move!
50) 47.9 mph [N 56.4 o W]
51) A quantity with both magnitude and direction.
23) 16.8 dynes [E 39o N]
24) 51 N [N]
52) 83.2 lb [E 57o N], 155 m/s [E 8.7o N], 293 km
[W 60o S]
25) 43.46 lb [E 77.5o S]
53) 0.57 m/s, 0.43 m/s [S 27o W]
26) 200 Newtons
54) 80 mph [W]
27) the sum is zero
55) 21.2 m/s/s [E 52o N]
28) 4500 N [W 20 S]
56) 25 lbs
29) 57.2o [W 53.5o N]
57) x-components must cancel; y-components must
cancel
30) 13 m/s [E]
31) 9.85 m/s[N 24o E]
58) 81.5 lb
59) 10.9 m [W 58o S]
60) 12 m up road (before the trashcan is encountered