CHAPTER 1
Introduction and
Mathematical Concepts
PREVIEW
In this chapter you will be introduced to the physical units most frequently encountered in physics. After completion
of the chapter you will be able to convert between these systems of units and use them as an aid in problem solving.
Also, after a review of trigonometry, you will become acquainted with vectors and the methods of vector
addition. Upon completion of this material you should be able to add and subtract vectors graphically, decompose
vectors into their components and use the components to reconstruct the vectors.
QUICK REFERENCE
Important terms
Scalar quantity
A quantity which can be described by a single number.
Vector quantity
A quantity which can be adequately described by a number (magnitude) and a direction.
Vector components
Two perpendicular vectors which added together produce the original vector.
Systems of Units
System
SI
BE
CGS
Length
meter (m)
foot (ft)
centimeters (cm)
Trigonometry
h
sin θ = o (1.1)
h
 ho
θ = sin-1  h 
(1.4)
ha
h
(1.2)
 ha
θ = cos-1  h 
(1.5)
(1.3)
ho
tan-1  h 
 a
cos θ =
h
tan θ = o
ha
θ=


(1.6)
Mass
kilogram (kg)
slug (sl)
gram (g)
Time
seconds (s)
seconds (s)
seconds (s)
h
ho
θ
ha
h2 = ho2 + ha2
(1.7)
Graphical Addition of Vectors
Vectors may be added (or subtracted) by placing them head to tail.
-B
B
R
A
A
R
R=A+B
R=A-B
Components of a Vector
The following expressions refer to the vector diagram shown below.
Ax = A cos θ
A =
2
Ay = A sin θ
2
Ax + A y
 Ay
θ = tan-1  A 
 x
y
Ay
A
θ
Ax
x
Addition of Vectors Using Components
If C = A + B, then the components of C are:
Cx = Ax + Bx
Cy = Ay + By
and the magnitude and direction of C are:
C=
2
2
Cx + Cy
where θ is measured counterclockwise from the + x axis.
 Cy
θ = tan-1  C 
 x
DISCUSSION OF SELECTED SECTIONS
1.3 The Role of Units in Problem Solving
Units are very important in the study of physics in that all physical quantities have units. These may be either the
base units of length (m), mass (kg), and time (s), or derived units such as the joule (kg m2/s 2).
When used in algebraic expressions, the units which accompany the numbers can be used to check not only the
accuracy of the calculation, but also the validity of the equation. For this reason, the units will always be displayed
along with the numbers in this study guide. You are encouraged to do the same in your solutions to problems.
Remember, if the units do not work out, your solution is not right either.
Example 1 Manipulating Units and Converting Between Systems
Convert 5.00 mi/h to m/s.
Conversion factors:
1 mi = 5280 ft
1 km = 1000 m
1 ft = 0.305 m
1 h = 3600 s
Each of the equalities above can be used to form a fraction or conversion factor that is equal to unity (i.e., equal to
1). In multiplying by unity we do not change the value of the physical quantity; we are merely expressing the same
quantity in a different set of units. One side of the equality will appear in the numerator and the other side will
appear in the denominator of the fraction. The specific choice will be made so that the unwanted units cancel and the
desired units appear in the final answer.
5.00 mi/h = 5.00
mi  5280 ft   0.305 m   1 h 
m



 = 2.24
h  1 mi  1 ft   3600 s 
s .
Notice that the first conversion factor changes miles to feet, the second changes feet to meters, and the third
conversion factor changes hours to seconds. Again notice that in each conversion factor, the numerator is equal to
the denominator and, therefore, the conversion factor equal to unity.
Example 2 Another Look at Converting Units
Convert 2.00 mi2 to ft2 .
We will start with the fact that 1 mi = 5280 ft. In setting up the conversion factor as a fraction equal to unity,
however, we must square the fraction so that it converts square miles (mi2 ) to square feet (ft2 ).
2
 5280 ft 
2.00 mi2 = 2.00 mi 2 
= 5.58 ×10 7 ft 2
 1 mi 
Notice that the conversion factor is still equal to unity (12 = 1). Also notice that the final answer is expressed in
scientific notation to preserve the correct number of significant figures; three in this case.
Example 3 Using Units to Check Equations
An equation which may result from the application of the conservation of energy principle is
1 mv2 = mgh
2
where m is a mass with the units of kg, v is a velocity with the units of m/s, g is an acceleration with the units of
m/s 2 and h is a height with the units of m.
This equation may be checked for validity by simply substituting the units into it and manipulating them as if they
were numbers. If the units on the right and left side of the equation do not match, the equation is definitely NOT
VALID.
( )
(kg)
( )
(kg)
( ) ( )
m
s
m2
s2
2
= (kg)
= (kg)
m (m)
s2
m2
s2
kg m2 kg m2
=
s2
s2
In this case, the units on each side of the equation do match and the equation may be a valid one. This procedure
does NOT guarantee that the equation has any "true" meaning, however.
By the way, the above combination of units appear so often in physics that they are given the special name of
joules (J). This is the unit of ENERGY.
1.4 Trigonometry
Most likely, you have already become acquainted with the basic trigonometric functions in a high school or college
course. These can be found in the Quick Reference section of this book.
Many times the use of the "trig" functions in physics involve finding the length of one side of a right triangle
when you know one other side and one of the acute angles. This is particularly important in finding the components
of a vector which will be studied in the next section.
Other variations are, of course, possible and useful. For instance, if two sides of a right triangle are known, then
the trigonometric functions can be used to find the angle.
Example 4 Finding one side of a right triangle if one other side and an angle are known
An observer, whose eyes are 6.0 ft above the ground, is standing 105 ft away from a tree. The ground is level, and
the tree is growing perpendicular to it. The observer's line of sight with the tree top makes an angle of 20.0° above
the horizontal. How tall is the tree?
The observer's eye, the tree top, and a point on the trunk are the vertices of a right triangle. The height of the tree is
H
= 6.0 ft + ho
Now ho can be found from the right triangle by using
equation (1.3)
h
ho
θ
H
ha
ho = ha tan θ = (105 ft) tan 20.0° = 38.2 ft.
H = 6.0 ft + 38.2 ft = 44.2 ft.
Another common need in physics is to find one side of a triangle when the other two sides are known but no acute
angle is given. Then you should use the Pythagorean theorem (1.7) or one of its variants.
Example 5 Using the Pythagorean theorem
In the previous example, it is desired to know the straight line distance from the person's eye to the top of the tree.
The Pythagorean theorem gives
h = h20 + h 2a = (105 ft)2 + (38.2 ft) 2 = 112 ft .
1.6 Vector Addition and Subtraction
If vectors are colinear, then they may be added or subtracted by simply adding or subtracting their magnitudes. The
directions of the vectors are usually specified by calling a vector pointing to the right (or up) positive and a vector
pointing to the left (or down) negative. Then the sign of the resultant vector tells which way it points.
Example 6 Adding and subtracting colinear vectors
Find the resultant, C , of the following vectors (u is an arbitrary unit).
Case 1
Case 2
A = +3.0 u
A = +3.0 u
B = +2.0 u
B = -2.0 u
In both cases the vectors are colinear. One could imagine placing the vectors A and B in a "tail-to-head" fashion. In
case 1, both A and B point in the same direction. The resultant vector will have a magnitude equal to the length of
A plus the length of B. Since both vectors are directed to the right, the resultant will also point to the right. In
case 2, the vectors A and B point in opposite directions. If they are placed in a tail-to-head fashion it will be clear
that part of A is canceled by B. The resultant will have a magnitude that is equal to the magnitude of A minus the
magnitude of B. Since A has the greater magnitude, the resultant will point in the direction of A. The resultant C
is shown below for both cases.
Case 1
Case 2
C = +5.0 u
C = +1.0 u
When vectors are not colinear but perpendicular, they may be added by using the Pythagorean theorem and the
tangent function. This method yields the magnitude and direction of the resultant vector as a number and an angle.
Example 7 Adding perpendicular vectors
Find the resultant, C, of the vectors shown where A = + 2.0 u and B = + 5.0 u.
The resultant is C = A + B as shown in the diagram.
Notice that it was constructed by placing the vectors A and B in a "tail-to-head" fashion and then connecting the
"head" of A to the "tail" of B.
The length of the resultant vector, C, is given by the Pythagorean theorem to be
2
2
C=
A +B
C=
(2.0 u) + (5.0 u)
2
2
C = 5.4 u.
C
A
θ
B
The direction of C, as specified by the angle, θ, is given by (1.6) to be
 A
 2.0 u 
o
θ = tan-1  B = tan-1  5.0 u  = 22 .
Now C
= 5.4 u, 22° counterclockwise from B.
1.7 The Components of a Vector
As you might infer from the preceding example, any vector, C, may be expressed as the sum of two perpendicular
vectors. When these perpendicular vectors are placed along the x and y axes of a Cartesian coordinate system, they are
referred to as the x and y components (Cx and Cy) of the vector.
The components of a vector may be found by using a vector diagram similar to the above and equations (1.1)
and (1.2). Please keep in mind that if equations (1.1) and (1.2) are used to find the components of a vector, the angle,
θ, is defined as being measured from the x axis.
Example 8 Finding the components of a vector
Find the components of the vector A = 5.0 u, θ = 30 °.
An application of (1.2) to the vector shown in the figure gives
Ax = A cos θ = (5.0 u) cos 30°
Ax = 4.3 u.
A similar application of (1.1) gives
Ay = A sin θ = (5.0 u) sin 30°
Ay = 2.5 u.
y
Ay
A
θ
Ax
x
Sometimes you will encounter situations where you will need to find the x and y components of a vector, but the
angle given, φ, is NOT measured from the x axis. In this case, the components may NOT be given by
Ax = A cos φ
and
Ay = A sin φ.
You now have two choices. You may either find the angle, φ, in terms of the angle, θ, or you may try to find a
different right triangle in which to apply equations (1.1) and (1.2).
Example 9 Finding the components of a vector when the angle is not measured from the +x axis
Find the x and y components of the vector, A, whose magnitude is 10.0 u and which makes an angle of 45° with the
+y axis. Refer to the figure shown.
We may use
Ax = A cos θ and Ay = A sin θ if we can find the angle, θ.
From the figure it is seen that
θ = 90° + φ so that
θ = 90° + 45° = 135°.
Ax
Ay
y
A
φ
θ
x
Now
Ax = (10.0 u) cos 135°
Ax = - 7.07 u
Ay = (10.0 u) sin 135°
Ay = + 7.07 u.
We may also apply equations (1.1) and (1.2) to the right triangle shown in the figure to obtain
Ax = - A sin φ = - (10.0 u) sin 45° = - 7.07 u
and
Ay = + A cos φ = + (10.0 u) cos 45° = + 7.07 u.
1.8 Addition of Vectors by Means of Components
Since like components of two or more vectors are colinear, they may simply be added, as in example 6, to give the
corresponding component of the resultant vector. In this way, all of the components of the resultant vector may be
found. The magnitude and direction of the resultant vector may then be determined by equations (1.3) and (1.7).
Example 10 Adding vectors by the component method
Add the vectors A, B, and C shown in the figure using the component method. A = 5.0 m, B = 7.0 m and
C = 4.0 m.
An application of equations (1.1) and (1.2) to each of the triangles shown in the figure gives
Ax = + A cos 20°
Ax = + (5.0 m) cos 20°
Ax = + 4.7 m.
Bx = - B cos 40°
Bx = - (7.0 m) cos 40°
Bx = - 5.4 m.
Cx = + C sin 25°
Cx = + (4.0 m) sin 25°
Cx = + 1.7 m.
y
B
A
40°
20°
x
C
25°
The x component of the resultant vector is
Rx = Ax + Bx + Cx
Rx = 4.7 m - 5.4 m + 1.7 m
Rx = 1.0 m.
Repeating the above for the y component gives
Ay = + A sin 20° = + (5.0 m) sin 20° = + 1.7 m.
By = + B sin 40° = + (7.0 m) sin 40° = + 4.5 m.
Cy = - C cos 25° = - (4.0 m) cos 25° = - 3.6 m.
The y component of the resultant vector is
Ry = Ay + By + Cy = 1.7 m + 4.5 m - 3.6 m
Ry = 2.6 m.
Now the magnitude of the resultant can be found from the Pythagorean theorem
2
2
R=
Rx + Ry
R=
(1.0 m) + (2.6 m) = 2.8 m
The angle that R makes with the +x axis is
2
2
θ = tan-1 (Ry/Rx) = tan-1 (2.6 m/1.0 m) = 69°.
PRACTICE PROBLEMS
The following problems are provided to give you additional practice solving single concept problems. Some work
space has been left for you after or to the right of the problem. The solutions to these problems will be found at the
end of this chapter.
1.
How many significant figures are in the numbers
a. 2.03
b. 0.0054
c. 2.8 X 105
d. 1500
e. 0.23 X 10-4
2.
Convert the following into the indicated units.
a. 5.00 m to ft
b. 1.2 ft/s to m/s
c. 60.0 mi/h to km/h
d. 9.80 m/s2 to ft/s2
e. 535 kg to slugs
3.
A basketball coach insists that his players be at least 180.0 cm tall. Would a player of height 5 ft 11.5 in tall
qualify for the team?
4.
A football field is 100.0 yards long. Express this distance in millimeters.
5.
Check the following equations for possible validity.
a. x 2 = 1/2 gt where x is in m, g is in m/s2, t is in s.
b. v 2 = 2 ax where v is in m/s, a is in m/s2 and x is in ft.
6.
A right triangle has a side of length 3.5 m. The angle opposite the 3.5 m side is 25°. Find the length of the
other sides.
7.
A 500.0 m tall building casts a shadow 800.0 m long over level ground. What is the sun's elevation angle
above the horizon?
8.
A right triangle has two sides of length 25 ft and 15 ft. Find the length of the hypotenuse and all angles.
9.
A bridge 50.0 m long crosses a chasm. If the bridge is inclined at an angle of 20.0° to the horizontal, what is
the difference in height between the two ends?
10. A displacement vector of magnitude 5.0 m points in an easterly direction. A second displacement vector points
north and has a magnitude of 9.7 m. Find the magnitude and direction of the vector sum.
11. An electric field vector, E, has a magnitude of 1.0 newtons per coul (N/C) and makes an angle of 33° CCW
from the +x axis of a Cartesian coordinate system. Find the components of E.
12. A magnetic field vector, B, is oriented 65° clockwise from the -y axis. It has a magnitude of 0.010 tesla (T).
Find the x and y components of B.
13. A vector, A, has a magnitude of 10.0 u and points 21.0° north of west. A second vector, B, has a magnitude of
5.2 u and points 47.0° east of south. Find the magnitude and direction of the sum by the component method.
14. A car drives 2 km west, then 8 km south, and then 10.0 km at an angle 53° north of east. Find the car's final
displacement. (magnitude and direction)
HELPFUL SUGGESTIONS
1.
When expressing a vector quantity for an answer, be sure to specify both its magnitude and direction.
2.
Vectors may be moved from place to place, providing that you maintain the same length and direction.
3.
Calculating the components of a vector as Ax = A cos θ and A y = A sin θ is correct only
measured with respect to the x axis.
4.
When giving the answer to a problem, ALWAYS include the units. The units can be used as a tool for checking
whether or not the answer is "dimensionally" correct.
5.
Try to avoid just "plugging in" numbers into equations. Try to understand the ideas and concepts rather than just
memorizing equations.
6.
When you obtain your solution, always ask yourself, "is the solution reasonable?", "does it make sense?", and
"are the units consistent?"
if θ is an angle
EVERYDAY PHYSICS
1.
If you want to lay out a garden, build a foundation or anything which should have 90 degree corners, use a 3-4-5
right triangle. Use a string to roughly define two adjacent sides of your garden, measure 3 ft along one side and 4
ft along the other side and mark the location of each point. Measure from mark to mark and adjust the angle
between the strings until the measurement yields exactly 5 ft. The strings will then be 90 degrees from each
other. For more accuracy you may want to use a 6-8-10 right triangle or even a 9-12-15 right triangle.
2.
Most of us are familiar with the SI units of meter and kilogram, but the British units of foot and pound are
deeply ingrained since they are the ones we use in everyday life. Become more familiar with the SI units by
lifting 1 kg of a common substance like sugar. Also, measure several common objects like your car in both
meters and feet.
3.
Determine the height of a tall object by measuring its shadow and estimating the angle of elevation of the sun.
If it is possible, find the actual height of the object and compare with your results. Can you think of ways to
determine the angle more accurately?
CHAPTER QUIZ
This quiz has been provided to help you diagnose possible weak areas in the understanding of the key chapter
concepts. The answers can be found at the end of the chapter in this guide.
1.
2.
3.
4.
How many significant figures are in the answer of 1.20 - 1.01?
a. one
b. two
c. three
d. four
How many significant figures are in the result of 1600/2.80?
a. one
b. two
c. three
d. four
The derived unit (kg2 m2)/(kg m s 2) is equivalent to
a. (kg2)/s2
b. (kg m/s2)
c. (kg m)/s
d. (kg m 2)/m
How many feet are in 25 m?
a. 82
c. 0.31
d. 3.28
b. 0.12
5.
If you are given one angle and the length of the side opposite in a right triangle, which trig function would
allow you, in a single step, to find the length of the hypotenuse?
a. sine
b. cosine
c. tangent
d. any
6.
If you are given one angle and the length of the hypotenuse of a right triangle, which trig function would you
use to find the side adjacent to the angle?
a. sine
b. cosine
c. tangent
d. any
7.
Vectors may be added graphically by placing them
a. tail to tail.
b. parallel.
c. tail to head.
d. head to head.
Which of the following is an example of a scalar?
a. force
b. volume
c. displacement
d. velocity
8.
9.
The parts of a vector which lie in perpendicular directions are called
a. magnitudes
b. resultants
c. components
d. scalars
10. If the angle, less than 90°, specifying the direction of a vector is given as measured from the y axis, then the x
component will ALWAYS involve what trig function?
a. sine
b. cosine
c. tangent
d. any
11. Which of the following is NOT a vector quantity?
a. time
b. velocity
c. force
d. displacement
12. Two displacement vectors have magnitudes of 8 m and 12 m , respectively. When these two vectors are added
the magnitude of the sum
a. is 20 m
c. is larger than 20 m.
b. is 4 m.
d. could be as large as 20 m or as small as 4 m.
13. A physical quantity is calculated from the formula K = 3π c (a2 + b2 ) where a, b, and c are all lengths.
What is the dimension of K?
a. [L]
c. 3π [L]2
2
b. [L]
d. [L]3
SOLUTIONS AND ANSWERS
Practice Problems
_____________________________________________________________________________________________
1.
a. 2.03 has THREE significant figure.
b. 0.0054 has TWO significant figures. Note that it can be written as 5.4 X10-3.
c. 2.8 X 105 has TWO significant figures.
d. 1500 has TWO significant figures since both zeros are in doubt.
e. 0.23 X 10-4 has TWO significant figures.
_____________________________________________________________________________________________
2.
a. We have
5.00 m = (5.00 m)(3.28 ft/1 m) = 1 6 . 4 f t .
b. Also
1.2 ft/s = (1.2 ft/s)(0.305 m/1 ft) = 0 . 3 7 m / s .
c. We know
60.0 mi/h = (60.0 mi/1 h)(5280 ft/1 mi)(0.305 m/1 ft)(1/1000 km/1 m)= 96.6 km/h.
d. In this case
9.80 m/s2 = (9.80 m/s2)(3.28 ft/1 m) = 3 2 . 1 f t / s 2 .
e. Finally
535 kg = (535 kg)(6.85 X 10-2 sl/1 kg) = 3 6 . 6 s l .
_____________________________________________________________________________________________
3.
The conversion looks like
180.0 cm = (180.0 cm)/(1 in/2.54 cm) = 70.9 in = 5 ft 10.9 in. YES
_____________________________________________________________________________________________
4.
In this case
100.0 yds = (100.0 yds)(3 ft/yd)(12 in/ft)(2.54 cm/in)(10 mm/cm) = 9 1 4 4 0 m m .
_____________________________________________________________________________________________
5.
a. Substitute the units into the equation to see if both sides match.
m2 = (m/s 2)(s) = m/s
The units on each side do NOT match so the equation is not valid.
b. Similarly,
m2/s 2 = (m/s 2)(m) = m2/s 2.
The units DO match so the equation MAY be valid.
_____________________________________________________________________________________________
6.
The hypotenuse is given by (1.1)
h = (3.5 m)/sin 25° = 8.3 m.
The side adjacent is given by (1.2),
ha = (8.3 m)cos 25° = 7 . 5 m .
_____________________________________________________________________________________________
7.
The shadow, the building, and a line drawn from the top of the building to the corresponding point on the end of
the shadow form a right triangle. The angle between this latter line and the horizontal is the elevation angle of
the sun and is
θ = tan-1 (500.0 m/800.0 m) = 32.01°.
_____________________________________________________________________________________________
8.
The length of the hypotenuse is given by (1.7),
h2 = (25 ft)2 + (15 ft)2 ,
h = 29 ft.
The angle opposite the 25 ft side is given by (1.6),
θ = tan-1 (25/15). θ = 59°.
The angle opposite the 15 ft side is
θ = tan-1 (15/25) = 31°.
Note that the angles add to 90° as they should.
_____________________________________________________________________________________________
9.
We are given the hypotenuse and one angle in a right triangle. (1.1) gives the side opposite the angle to be
ho = (50.0 m) sin 20.0° = 1 7 . 1 m .
_____________________________________________________________________________________________
10. The vectors are perpendicular; hence the magnitude of the vector sum can be found from the Pythagorean
theorem.
magnitude = 1 1 m .
The angle the vector makes with the east-west line (+x) is
θ = tan-1 (9.7/5.0) = 63° N of E.
_____________________________________________________________________________________________
11. The x component is
Ex = (1.0 N/C) cos 33° = 0 . 8 4 N / C .
The y component is
Ey = (1.0 N/C) sin 33° = 0 . 5 4 N / C .
_____________________________________________________________________________________________
12. The angle as measured from the +x axis is θ = 270° - 65° = 205°.
Then
Bx = (0.010 T) cos 205° = - 0 . 0 0 9 1 T
and
By = (0.010 T) sin 205° = - 0 . 0 0 4 2 T .
_____________________________________________________________________________________________
13. The x components are:
Ax = -(10.0 u) cos 21.0° = -9.34 u, Bx = (5.2 u) sin 47.0° = 3.80 u, Rx = - 5.54 u.
The y components are:
Ay = (10.0 u) sin 21.0° = 3.58 u,
By = -(5.2 u) cos 47.0° = -3.55 u, Ry = 0.03 u.
The magnitude of the resultant vector is found from the Pythagorean theorem to be
R =
2
(- 5.54 u) + (0.03 u)
2
= 5.54 u.
The angle is
θ = tan-1 (0.03/5.54) = 0 . 3 1 0 ° N o f W .
_____________________________________________________________________________________________
14. Let A be the westerly displacements, B the southerly displacement, and C be the remaining displacement.
Taking east and north to be positive, the east-west (x) components are:
Ax = -2 km, Bx = 0 km,
C x = +(10.0 km) cos 53° = 6 km.
R x = +4 km.
The north-south (y) components are:
Ay = 0 km,
By = -8 km,
C y = +(10.0 km) sin 53° = +8 km.
R y = 0 km.
The magnitude of the resultant vector is given by the Pythagorean theorem to be
R =
2
(4 km) + (0 km)
2
= 4 km.
and the direction is
θ = tan-1 (0/4) = 0°. That is, EAST.
_____________________________________________________________________________________________
Quiz answers
1. b
2. b
3. b
4. a
5. a
6. b
7. c
8. b
9. c
10. a
11. a
12. d
13. d
MCAT REVIEW PROBLEMS
_____________________________________________________________________________________________
When an airplane flies, its total velocity with respect to the ground is
v total = v plane + v wind,
where v plane denotes the plane’s velocity through motionless air, and v wind denotes the wind’s velocity. Crucially,
all the quantities in this equation are vectors. The magnitude of a velocity vector is often called the “speed.”
Consider an airplane whose speed through motionless air is 100 meters per second (m/s). To reach its
destination, the plane must fly east.
The “heading” of a plane is the direction in which the nose of the plane points. So, it is the direction in
which the engines propel the plane.
______________________________________________________________________________
_______________
1.
If the plane has an eastward heading, and a 20 m/s wind blows towards the southwest, then the
plane’s speed is:
A.
80 m/s
B.
more than 80 m/s but less than 100 m/s
C.
100 m/s
D.
more than 100 m/s
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2.
The pilot maintains an eastward heading while a 20 m/s wind blows northward. The plane’s
velocity is deflected
from due east by what angle?
A.
20
sin -1 100
B.
20
cos-1 100
C.
20
tan-1 100
D.
none of the above
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3.
Let φ denote the answer to question 2. The plane in question 2 has what speed with respect to the ground?
A.
C.
(100 m/s)sin φ
100 m/s
sin φ
B.
D.
(100 m/s)cos φ
100 m/s
cos φ
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4.
Because the 20 m/s northward wind persists, the pilot adjusts the heading so that the plane’s total velocity is
eastward. By what angle does the new heading differ from due east?
A.
20
-1
100
sin
B.
20
-1
100
cos
C.
20
-1
100
tan
D.
none of the above
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5.
Let θ denote the answer to question 4. What is the total speed, with respect to the ground, of the plane in
question 4?
A.
(100 m/s)sin θ
B.
(100 m/s)cos θ
C.
100 m/s
sin θ
D.
100 m/s
cos θ
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ANSWERS TO MCAT REVIEW PROBLEMS
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1.
B. In all of these problems, a good diagram gets you half way to the answer. For this plane, v plane = 100
m/s eastward, and v wind = 20 m/s towards the southwest. To represent the total velocity, add these two
vectors:
FIGURE 1
Just by looking at this rough diagram, you can see that the wind slows down the plane. So, the
answer must be A or B. Since the wind does not blow westward, it does not directly oppose the
plane’s motion. Therefore, the wind cannot slow the plane down by a full 20 m/s from the plane’s
“default” velocity of 100 m/s.
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2.
C . Because the pilot maintains an eastward heading, the engines still push the plane due east. So, v plane =
100 m/s eastward. But the wind blows at v wind = 20 m/s northward, thereby knocking the plane off course
by angle φ.
FIGURE 2
In this triangle, we know the two legs but not the hypotenuse. Specifically, in units of m/s, the
“opposite” leg has length ho = 20, while the “adjacent” leg has length ha = 100. Therefore, by the
definition of tangent,
ho
20
tan φ = ha = 100 .
20
Take the inverse tangent of both sides to get φ = tan-1 100 .
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3.
D . By looking at figure 2, and remembering the definition of cosine, we get
v plane
ha
100 m/s
cos φ = h = vtotal = vtotal .
Multiply through by v total, and then divide through by cos φ, to get
100 m/s
v total = cos φ .
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4.
A . The pilot now adjusts the heading so that v total points due east. Since the wind knocks the plane
northward, the plane must head slightly southward, to “offset” the wind:
[--- Unable To Translate Graphic ---]
FIGURE 3
Here, v plane is the hypotenuse. From figure 3, we get
sin θ =
ho
h
=
v wind
20 m / s
=
,
vplane
100 m / s
20
and hence, θ = sin -1 100 .
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5.
B. Again from figure 3, the hypotenuse has length h = v plane = 100 m/s, while the adjacent leg has length
ha = v total . So,
cos θ =
ha
vtotal
v total
=
=
.
h
v plane
100 m / s
Multiply through by 100 m/s to get
(100 m/s)cos θ = v total .
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