Download Chapter 3. Vectors

Chapter 3. Vectors
I. Vectors and Scalars
1. What type of quantity does the odometer of a car measure?
a) vector;
b) scalar;
c) neither scalar nor vector;
d) both scalar and vector.
2. What type of quantity does the speedometer of a car measure?
a) vector;
b) scalar;
c) neither scalar not vector;
d) both scalar and vector.
3. Which of the following is a vector?
a) mass;
b) temperature;
c) speed;
d) acceleration;
e) time.
4. During baseball practice, a batter hits a very high fly ball, and then runs in a
straight line and catches it. When comparing the displacement for the two, one
sees that the __________?
a) player's displacement is larger;
b) ball's displacement is larger;
c) two displacements are equal.
5. Which of the following is a scalar?
a) mass;
b) displacement;
c) velocity;
d) acceleration;
e) force.
II. Components of Vectors
1. Convert α = 54° to radians.
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2. Convert β = π to degrees.
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r
3. Find the iˆ ( ĵ ) component of the vector a if its magnitude is 15 m and the angle
θ that the vector forms with iˆ is 15o.
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4. An airplane is taking off with the speed of 600 km/h. Its path is linear, and it
makes an angle of 17o with the horizontal. Assuming that the speed of the airplane
remains constant, find the horizontal distance that the airplane travels in 3.5 s.
5. A 12-meter ladder that stands against a wall makes an angle of 50o with the floor.
If a worker gets all the way on top of the ladder, what is the vertical distance he
travels?
r
r
r
6. Two vectors a and b are given: a = (1.2m)iˆ + (3.0m) ˆj and
r
r r r
b = (−1.0m)iˆ + (2.7 m) ĵ . Find the magnitude of c = a + b .
r
r
r
7. Such vectors a and b are given that a = (4.0m)iˆ + (6.3m) ˆj and
r
r r r
b = (6.0m)iˆ + (4.9m) ĵ . Find the direction of vector c = a − b in degrees relative
to iˆ .
III. Adding and Subtracting Vectors Graphically
1. Two vectors A and B are given. What would be the resultant vector C if A and B
are added together?
a)
b)
c)
e)
d)
Hint 1/Comment:
Vectors can be added graphically without decomposing the vectors into vertical
and horizontal components.
Comment:
In order to add the two vectors graphically, place the tail of the vector B at the
head of the vector A. The resultant vector (vector C) can be found by drawing a
vector from the tail of vector A to the head of vector B.
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2. Two vectors A and B are given. Find C = A – B.
a)
c)
e)
d)
b)
Hint 1/Comment:
Vectors can be added graphically without decomposing the vectors into vertical and
horizontal components.
Hint 2/Comment:
In order to subtract the two vectors graphically, treat the subtraction as addition of a
negative vector A – B = A + (-B). The negative vector can be represented as an
arrow with the same magnitude as the original but pointing in the opposite direction.
Comment:
Place the tail of the vector –B at the head of the vector A. The resultant vector
(vector C) can be found by drawing a vector from the tail of the vector A to the head
of the vector B.
3. Given three vectors A, B and C, find D = A + B + C.
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a)
c)
e)
b)
d)
Hint 1/Comment:
Vectors can be added graphically without decomposing the vectors into vertical
and horizontal components.
Comment:
In order to add the three vectors graphically, place the tail of the vector B at the
head of the vector A, then place the tail of the vector C at the head of the vector
B. The resultant vector (vector D) can be found by drawing a vector from the tail
of vector A to the head of vector C.
4. Given three vectors A, B and C, find D = A - B + C.
a)
c)
b)
d)
e)
Hint 1/Comment:
Vectors can be added graphically without decomposing the vectors into vertical
and horizontal components.
Hint 2/Comment:
In order to subtract two vectors graphically, treat the subtraction as addition of a
negative vector A – B = A + (-B). The negative vector can be represented as an
arrow with the same magnitude as the original but pointing in the opposite
direction.
Comment:
In order to find D = A - B + C, place the tail of the vector –B at the head of the
vector A, then place the tail of the vector C at the head of the vector –B. The
resultant vector (vector D) can be found by drawing a vector from the tail of
vector A to the head of vector C.
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5. Given four vectors A, B, C and D, find E = 2A + 2B + C + D.
a)
c)
e)
b)
d)
Hint 1/Comment:
Vectors can be added graphically without decomposing the vectors into vertical
and horizontal components.
Hint 2/Comment:
Vectors 2A and 2B can be represented graphically as arrows having twice the
length of the respective original vectors (vectors A and B).
Comment:
In order to find E = 2A + 2B + C + D, place the tail of the vector 2B at the head
of the vector 2A, then place the tail of the vector C at the head of the vector 2B.
After that place the tail of the vector D at the head of the vector C. The resultant
vector (vector E) can be found by drawing a vector from the tail of vector A to the
head of vector D.
IV. Position, Displacement and Velocity Vectors.
1. Find the magnitude of the displacement of a truck (car, motorcycle) that traveled
130 mi east and then 50 mi west. Assume that the x-axis is directed east.
Hint 1/Comment:
Given the direction of motion and the distance that the object moved in that
direction, draw two collinear vectors that describe the motion.
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Comment:
Suppose that the vector A illustrates the object’s motion eastward, and the vector
B illustrates the motion westward. The picture below shows vector C as the sum
of vectors A and B (assuming that the magnitude of the vector A is greater than
that of the vector B).
2. An airplane is flying straight north with the speed of 560 km/h. The wind that
begins to blow east (from the west) has a speed of 35 km/h. Find the resultant
speed (the direction of the resultant velocity vector relative to east) of the
airplane. Assume that the x-axis is directed east, and the y-axis is directed north.
Hint 1/Comment:
Given the magnitudes and directions of the vectors, you can draw the described
vectors.
Hint 2/Comment:
Suppose that vector A represents the velocity of the airplane and vector B
represents the velocity of the wind.
The picture below shows the resultant vector (vector C).
Note that the three vectors form a right triangle.
Comment:
Use Pythagorean theorem to find the magnitude of the resultant velocity vector
(vector C).
Angle θ specifies the direction of the resultant velocity vector of the airplane
relative to east.
θ = 90o – α, where α = tan-1(B/A).
3. A camp of rock climbers is located at point A. Early in the morning, the rock
climbers travel 500 m to point B, the base of the cliff, then they climb straight up
for 100 m and reach point C. What is the degree measure of angle θ , the angle
r
that the displacement vector s makes with the horizontal?
Comment:
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Use trigonometric functions to find the direction of the resultant displacement
vector S: θ = tan-1(BC/AB).
4. A motorcycle is ridden (car is driven) 12 km north, 15 km west, and then 4 km
north. What is the magnitude of the displacement from the point of origin?
Hint 1/Comment:
Given the magnitudes and directions of the vectors, you can draw the described
vectors.
Hint 2/Comment:
Add the two vectors in the north direction together and treat them as one vector.
Comment:
Suppose that vector A represents the two vectors directed north and that vector B
represents the displacement vector directed west. Then C is the resultant
displacement vector (derived by adding vectors A and B graphically).
Vectors A, B, and C form a right triangle, so Pythagorean theorem can be used to
find the magnitude of vector C.
5. A car is driven (motorcycle is ridden) for 34 km south, then west for 15 km, and,
finally, 9 km in a direction 26o east of north. Find the magnitude of the car’s total
displacement.
Hint 1/Comment:
Decompose the vectors into east-west and north-south components.
Hint 2/Comment:
Add or subtract the components as necessary in order to obtain two vectors: one
in north-south direction and one in east-west direction.
Comment:
Adding these two vectors together gives a resultant vector, then the three vectors
form a right triangle.
Use Pythagorean theorem to find the magnitude of the resultant displacement
vector.
6. The velocity vector of a particle is equal to v = (−11.2m / s)iˆ − (15.5m / s ) ˆj . What
is the magnitude of this vector in m/s?
Hint 1/Comment:
The velocity vector is already decomposed into i and j components.
Hint 2/Comment:
Adding the two components graphically will give a resultant vector. The three
vectors then will form a right triangle to which Pythagorean theorem can be
applied to find the magnitude of the resultant vector.
V. Relative Motion
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1. The speed of a bicyclist that is riding in the direction of the wind is 10 km/h
relative to the wind. If the speed of the wind is 7 km/h relative to the ground, what
is the speed of the bicyclist relative to the ground?
Comment:
The velocity of the bicyclist relative to the ground is equal to the velocity of the
bicyclist relative to the wind plus the velocity of the wind relative to the ground:
vBG = vBW + vWG.
2. John (Jacob, Dave, Bryan, Chris, Terry, Kevin, Joe, Mike) runs at 10 km/h
relative to the ground. He throws a rock (stone, object) in the opposite direction
with the speed of 10 km/h relative to him. What is the velocity of the rock relative
to the ground? (Assume that John is moving in the positive direction of the xaxis).
Comment:
The velocity of the rock relative to the ground is equal to the velocity of the rock
relative to the boy plus the velocity of the boy relative to the ground: vRG = vRB +
vBG. Note that John throws the rock in the opposite direction he is running, so the
velocity of the object relative to the boy is negative.
3. The speed of each of the two trains that are approaching each other is 90 km/h
relative to the ground. What is the speed of one of the trains relative to the other?
Comment:
The velocity of the first train relative to the second train is equal to the velocity of
the first train relative to the ground plus the velocity of the second train relative to
the ground: vT1T2 = vT1G + v T2G.
4. A boat is traveling downstream at 20 km/h with respect to the water. If a person
on the boat walks from front to rear at 3 km/h with respect to the boat and the
speed of the water is 5 km/h relative to the ground, what is the velocity of the
person on the boat relative to the ground? Assume that the water is moving in the
positive direction of the x-axis.
Comment:
Step 1:
First we will find the velocity of the person relative to the water.
The velocity of the person relative to the water is equal to the velocity of the
person relative to the boat plus the velocity of the boat relative to the water: vPW
= vPB + vBW. You will use this velocity (vPW) in Step 2. Note that the boat is
moving downstream and the person walks from the front of the boat to the rear
i.e. in the opposite direction. This means that the velocity of the person relative to
the boat is negative.
Step 2:
Now we will find the velocity of the person relative to the ground.
The velocity of the person relative to the ground is equal to the velocity of the
person relative to the water (from Step 1) plus the velocity of the water relative to
the ground: vPG = vPW + vWG.
5. You are swimming across the river at 4 km/h relative to the water. The speed of
the water is 2 km/h relative to the ground. What should be the direction of your
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velocity relative to the water in order for your velocity vector with respect to the
ground to be perpendicular to the shore? (Give the measure of the angle that is
adjacent to the line perpendicular to the shore. State your answer in deg).
Hint 1/Comment:
The velocity of the person relative to the ground equals to the velocity of the
person relative to the water plus the velocity of the water relative to the ground:
vPG = vPW + vWG.
The graphical vector addition forms a right triangle. Note that α is the angle we
are asked to find.
Comment:
sin α = vWG / vPW, thus α = sin-1(vWG / vPW)
6. The boat is traveling upstream on a river. The speed of the water relative to the
ground is 2 m/s. What is the speed of the boat relative to the water if its velocity
relative to the ground is 16 m/s an angle of 10o upstream?
Hint 1/Comment:
The velocity of the boat relative to the ground equals to the velocity of the boat
relative to the water plus the velocity of the water relative to the ground: vBG =
vBW + vWG (see picture), where angle α = 10o (according to the text of the
problem).
Hint 2/Comment:
We need to find the velocity of the boat relative to the water (vBW). Thus, vBW =
vBG – vWG.
Use trigonometric functions to break vBG into horizontal (x-) and vertical (y-)
components, and then subtract respective components of the vectors vBG and vWG.
(Note that vWG is directed in the opposite direction of the vertical component of
vBG, so assuming that the vertical component of vBG is positive, the vertical
component of vector vWG is negative).
Comment:
The horizontal component of vector vBG equals: (vBG)x = vBG cos α. The vertical
component of vBG equals: (vBG)y = vBG sin α.
The horizontal component of vector vWG equals: (vWG)x = 0. The vertical
component of vWG equals: (vWG)y = - vWG.
Thus, in unit vector form, vector vBW is represented as follows: vBW = (vBG cos α)
i + (vBG sin α) j - (- vWG) j = (vBG cos α) i + (vBG sin α + vWG) j.
Use Pythagorean theorem to find the magnitude of the vector vBW.
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