AP PHYSICS C

SUMMER

ASSIGNMENT

Ax= A cos 9 sin 9 = OPP

HYP

Ay =A sin 9

a = b = c sin A sin B sin C

cos 9 = ADJ HYP

tan 9 = OPP

ADJ

AP PHYSICS C SUMMER ASSIGNMENT

The summer assignment for AP Physics C includes a review of vector algebra in two

dimensions and a review of calculus concepts involving basic derivatives. Your

assignment will be collected, checked and graded within the first week of school (it

should be ready to be handed in the first day!). Follow the directions provided to make

sure you are answering the questions completely.

The vector review includes:

• vector components

= the components method

• polar coordinate systems

The calculus review includes derivatives using the:

• power rule

• product rule

• quotient rule

Directions:

• Do all calculations and diagrams in pencil.

Your summer assignment grade will take into account neatness therefore you

should do all your work in pencil so you can erase any mistakes. Work that has

cross-outs or is sloppy and/or illegible will lose points.

• When requested, diagrams should be drawn and labeled neatly. Diagrams

should be large enough so that all angles and vectors are clearly seen. A typical

vector diagram should be at least 3 inches in height. (It is better to make a

diagram larger than necessary rather than too small.) All vector diagrams should

have the vector magnitudes and appropriate angles labeled. Be sure to include

arrows on all vectors! Do not include vector diagrams where they are not

requested, such as in the components method section.

• Solutions to the problems should include the equation(s) used, the

substitution step (where the numbers are plugged in) and the answer.

Units must be on all steps of the solution, not just the answers! Any solutions that do not follow this exact procedure (which is standard on both the AP test and

the regents test) will lose points. Check your answers to the blue packet with the

answer key provided.

• Do not worry about significant figures.

However, a good rule to remember is that your answer should be taken out 1 or 2

places after the decimal. Answers that are not accurate enough or are taken out

too many decimal places can lose points on the AP exam.

• Rounding is taken into account.

If your answer does not exactly match the answer key, but is close, it is probably

correct. There are many different ways to get to the solution to a problem and

rounding may change the answer slightly. If your answer is significantly off from

the answer key, you may want to check your work again or contact me.

• For the calculus problems, all steps in determining the derivative should be

shown, not just the final answers. Answers should be simplified and reduced, if

possible. For problems that involve evaluating the derivative at a specific value

of x, include the substitution step in your solution.

Once school begins, some class time will be devoted to answering any questions you may have on the summer assignment, so as you are doing the problems, make note of any questions you may want to go over or concepts you need clarified.

It is definitely advisable to work on the summer assignment in groups so that you may

learn from each other and help teach each other as well. However, there is a difference

between "working together" and copying someone else's work. Since all students will

be evaluated independently once classes begin, it is advisable that everyone work to

the best of his/her abilities on the summer assignment (and during the course of the

year). If you do not do your own work, it will surely show when the first test is

administered.

The summer assignment should be ready to hand in the first day of school! Any

assignments that are submitted after the due date will lose 10 points per day. Summer

assignments will not be accepted after the corrected assignments are handed back. If

you have any questions about the summer assignment, you can contact me over the

summer at jhanley@wvcsd.org after July 15th.

Enjoy the summer!

VECTORS AND SCALARS

VOCABULARY:

SCALAR· quantity expressed in terms of magnitude/ size only, with units.

EX) time

mass

distance

speed

work

power

energy

VECTOR • quantity expressed in terms of magnitude (with units) and direction.

EX) displacement

velocity

force

acceleration

momentum

impulse

torque

RESULTANT • single vector which equals the effect of all the individual vectors conmbined.

TIP·TO·TAIL RULE • to determine the direction of the resultant, arrange all vectors (without changing direction) so that the tip of one vector touches the tail of another. The resultant is then drawn from the free tail to the free tip.

EQUILIBRIUM • state of BALANCE in which the net/resultant force acting on an object equals zero.

EQUILIBRANT - force which is equal in magnitude but opposite in direction to the resultant force; produces equilibrium.

A

8

B

RULES: VECTOR ALGEBRA REVIEW

1. Vectors at oo = same direction: ADD (maximum resultant)

• > > • A

2. Vectors at 180° =opposite direction/ antiparallel: SUBTRACT (minimum resultant)

< A • )

The angles of oo and 180° are very important because since they represent the angles at which the largest {maximum) and smallest (minimum) resultants occur, all other angles must )yield resultants which fall between the maximum and minimum values.

3. Vectors at right angles (90°): use the Pythagorean theorem & SOHCAHTOA

sine= OPP HYP

cos8= ADJ HYP

tan 8 = OPP

ADJ

4. Vectors at "other" angles: use Laws of sines and cosines:

c2 = a2 + b2 - (2ab cos C)

a

= b = _c_

sin A sin B sin C

- ....

PROPERTIES OF VECTORS

1. Muitipiying a vector by a scaiar:

kA is a vector that has the same direction as A, but has a magnitude multiplied by value "k".

2. Negative Vectors:

A negative vector, ·A, has the same magnitude as vector A, but is in the opposite direction.

3. Vector Addition (Vector Sum):

A + B = C refers to the process in which a resultant (or unknown vector) is found by using the appropriate mathematical method (addition, Pythagorean theorem, law of sines/cosines etc) or solving graphically.

4. Vector Subtraction:

Vector subtraction can be treated as vector addition with a negative vector.

_... .... .... A· B =A + (·B)

5. Commutative Law of Addition:

The vector sum of two vectors is independent of the order in which they are combined.

_. ...... .. .

A+B=B+A

6. Associative Law of Addition:

The vector sum of three or more vectors is independent of how the individual vectors are grouped together.

..... ...... _.. ...... ..... .

A + (B + C) = (A + B) + C

VECTOR COMPONENTS

Components - l!ldependent vectors parallel to the x or y axis that together for the original vector, A.

Vector Resolution - the process of determining the components of a wctor. If a vector is

denoted A, then the two perpendicular vector components would be denoted Ax and Ay.

Ax

To find the magnitudes of Ax and Ay, the following equations can be used:

Ax = Acos8

Ay = A sin 8

where A is the magnitude of the resultant/hypotenuse and 8 is the angle between the resultant/hypotenuse and the x axis.

1. A 52 N force is applied to a particle, as shown below. The vertical component of the force is 18 N. Draw and label a vector diagram and calculate the magnitude of the horizontal component of the applied force and the angle, e, the force makes with the horizontal.

2. A missile is launched with a velocity of 860 m/s at 27° north of west. Draw and label a vector diagram and calculate the magnitudes of the northern and western components of this velocity.

N

3. The diagram below shows a child sitting on a sled on a snowy slope inclined at an angle of 14°, relati' ·o to the horizontal. The mass of the child-sled sy,tsm is 48.0 Kg. Draw and label a vector diagram and calculate the components of the child-sled system's weight parallel and perpendicular to the incline.

THE COMPONENTS METHOD

The components method is a method of vector algebra that can be used for any vector problem. However, due to it's time consuming nature, it is best used for problems that contain more than two vectors and/or

have angles other than oo, 180° or 90°.

COMPONENTS METHOD CHART:

9 is measured counterclockwise from the positive x axis!

l Acos8 Asin8

Vector 8 X y

A Ax Ay

B Bx By

c Cx Cy

D Dx Oy

Rx=Ax+Bx+Cx+ ...

Ry= Ay + By+ Cy + ...

R = JRx + Ry

tan 8 = Ry/Rx

(or a2 + b2 = c2)

1. Three forces act concurrently on an object: F1 = 26 N, 18° west of north

F2 = 40 N, 35° south of east

F3 =57 N, 20° south of west

Determine the magnitude and direction (9) of the resultant force and the equilibrant force, which will produce equilibrium.

-

-

2. Given forces, A, B and C:

A= 10 N, 22° north of east

B= 7 N, 40° north of west

C = 19 N, 16° east of south

_. _. - Determine the magnitude and direction (9) of -A + 28 + C.

jJI

3. A rabbit hops 1.5 meters west, 1.0 meter north, 2.0 meters west, 4.2 meters south and then 9.6 meters east. Determine the magnitude and direction (8) of the rabbit's resultant rUl;s"'1a"vei'YIl'lIeInt ,

-

4a. Three forces act concurrently on a particle. The magnitude and direction of each force is:

.A...

= 35 N, 20° north of west

8 = 18 N, 15° west of south _..

C = 21 N, 35° south of east

Determine the magnitude and direction

_. ..... (8) of A - 38 + C.

-

..... ......

b. Determine the magnitude and direction (9} of- 2A + B - C.

5. A person hikes 5.6 Km at 38° north of west, then 6.0 Km at 10° north of east, then 13.5 Km at 52° south of west. The person ends up 4.3 Km, 25° south of east of her starting point. Determine the magnitude and direction (8) of the fourth part of her hike.

6. Three forces are located in the xy plane and act concurrently on a proton. Force A has _... magnitude of 62.0 N d is directed at 30° above the -x axis. The m nitudes of forces B and C are unknown. Force B is directed at 65° below the -x axis and force C is directed at r above the +x axis. The three forces togeth resu!in the proton being in equilibrium. Algebraically determine the magnitudes of forces B and C.

POLAR COORDINATE SYSTEMS

To represent the location of a point in space, one can use polar coordinates (r, 9) where r is the distance from the origin to a point with rectangular/Cartesian coordinates (x, y) and 9 is the angle between the line (f;-om the origin to the point) and a fixed axis (usual!j the positive x axis, so angles could be measurer' counterclockwise from this axis).

y

(x,y)

EXAMPLES: Determine the Cartesian coordinates of the following points expressed in polar coordinates:

X-= rcos e X=lfocos/ 0°

X= -15. O't

f.= rcos e

X =- 2 g cos 7oo

= r-sine • 0

y=lbSit'\1 0

'I -= 5. 5

y= rsi" e

'/ = 2 8 sin 70o

'I= 2lo.

c. (9, 325°)

X= rcos e X=- 'f cos 325°

X= 7.'f

d. (31, 220°)

X::: rcos e X= 3\ tos 220°

X= -23.7

y= rsifle

'I= 'f.s ,·Y\ 325 °

= -.5.2

'/= rsir'\9

y::::. 3\ sin 220°

y::::.- lq.9

1. Determine the Cartesian coordirates of the following points:

2. A point has Cartesian coordinates (x, 4) and polar coordinates (r, 42°). Determine the values of x and r.

3. Point A has rectangular coordinates (5, 8). Point 8 has rectangular coordinates (-2, -4).

y .A

--+-t --1-1- +--+--+--+-+-r--r-- 1--+-+-+--+--+--t-+ --.-- X

•

a. What are the polar coordinates of each point?

b. Calculate the di tance between point A and point B.

4. A point has rectangular coordinates {-5.6, y) and polar coordinates {r, 230°). Determine the

vales of y and r.

5. Point Pis located in the 3rd quadrant and has rectangular coordinates {x, -15.5) and polar coordinates {26.8, e). Determine the values of x and e.

6. Point Q is located in the 4th quadrant and has Cartesian coordinates (8.3, y) and polar

coordinates (13.2, e). Determine the values of y and e.

THE DERIVATIVE

The derivative of a function represents the slope of the tangent line at a given point and is equal to:

lim h --+0

f(x + h) • f (x)

h

y

X X+h

The slope of the secant line represents an average value (such as average velocity, average acceleration). Ash approaches zero, the slope becomes a tangent line, which represents an instantaneous value (such as instantaneous velocity or acceleration).

RULES FOR FINDING DERIVATIVES:

A. Power Rule

y = Axn dy/dx = n A x" ·1

B. Product Rule

d/dx [ f(x) g(x) ] = f(x) d/dx [ g(x) ] + g(x) d/dx (f(x)]

C. Quotient Rule

d/dx [f(x)/g(x)] = g(x) d/dx [f(x)] • f(x) d/dx [g(x)] [g(x)]2

D. Trig functions

y = A sin (ax) dy/dx = A a cos (ax)

y = A cos (ax) dy/dx = ·A a sin (a x)

E. Chain Rule

df = df • du dx du dx

F. Functions with e

y = eax dy/dx = a eax

G. Functions with In

f(x) = u y = AInu dy/dx = A (1/u) du/dx

A. The Power Rule: y = Axn dy/dx = n A xn-1

For each of the following, find dy/dx (or y') and simplify/reduce answers, if possible.

1. y = 3x6 + 4x - 2/x2 5. y = 4x5 + x4 + 8

2. y = 5/x3 + 2x2 + 8 {X 6. y = l...fX4 - 6x-3

3. y = 6x113 + \fX3 - TT 7. y = 1/4 xs - 3/4 x6 + {X

4. y = 3/2 xS - 2/5 x115 8. y = -3/x4 + 10x-2

9. If y = 5x4 - 15x2 + 20, evaluate dy/dx at x = -3.

10. If y = 8/fX + 3fy2, evaluate dy/dx at x = 4.

11. If y = -4x3 + 2x2 - 12x, evaluate dy/dx at x = -2.

12. If y = 2/3 x3 + 3 x2 - 8 x, at what value(s) of xis dy/dx equal to zero?

13. If y = 1/3 x3- x2- 24 x + 6, at what value(s) of xis dy/dx equal to zero?

B. The Product Rule: d/dx [ f(x) g(x) ] = f(x) d/dx [ g(x) ] + g(x) d/dx (f(x)]

In other words, the first function times the derivative of the second function + the second function times the derivative of the first function.

For each of the following, find dy/dx (or y') and simplify/reduce answers, if possible.

1. y = (2x3 + 6x) (4x2- 5x)

2. y = (3x + 1)(x2 + 6x - 2)

3. y = (6; 2 - 2)(4-fX + 1)

4. y = (x3 - 2x + 3){2 + 3/x)

5. y = {xs + 4x) (6Jx2 - x)

6. y = (-2x4 + 6x3 + 3x + 5) {2x2 - 4)

7. If y = (5x + 7) (x3 -10), evaluate dy/dx at x = 5.

8. If y = (-4x2 + 12x) (2x3- 8), evaluate dy/dx at x = -2.

9. If y = (x2 + 5) (x4- 4), evaluate dy/dx at x = 1/2.

10. If y = (x4- 3x) (4x2 + 2), evaluate dy/dx at x = 3.

C. The Quotient Rule: d/dx [f(x)/g(x)] = g(x) d/dx [f(x)] • f(xl d/dx [g(x)] [g(x))2

In other words, the bottom function times the derivative of the top function minus the top function times the derivat;ve of the bottom function divided by the bc..tom function squared.

For each of the following, find dy/dx (or y) and simplify/reduce answers, if possible.

1. y = 2x2 + x - 6

X + 2

2. y = 3x

2 + x2

4. y = 2x2 - 4

X

5. y = 4x2 + 6 2x

6. y = 4x

2x -1

7. y = 3x3 + 8

6-x2

8. If y = 5x2 evaluate dy/dx at x = 5.

3- x2

9. If y = 10x evaluate dy/dx at x = 3. 2x- 3

10. If y = 2x2 - 8 evaluate dy/dx at x = 0.

x+4

11. If y = 3x3 -12 evaluate dy/dx at x = -1. 2x2 + 1

ANSWER KEY

Vector Components:

1. 48.79 N, 9 = 20.3°

2. Western component = 766.27 m/s, Northern component = 390.43 m/s

3. F = 113.8 N, F = 456.43 N

Components Method:

1. R = 33.8 N, 9 = 31.6° south of west (or 211.6° cc)

E = 33.8 N, 9 = 31.6° north of east (or 31.6° cc)

2. 19.7 N, 9 = 41.3° south of west (or 221.3° cc)

3. 6.9 m, 9 = 27.r south of east (or 332.3° cc)

4a. 52.11 N, 9 = 88.1o north of west (or 91.9° cc)

b. 52.79 N, 9 = 33.r south of east (or 326.3° cc)

5. 11.55 Km, 9 = 22.2° n'Jrth of east (or 22.2° cc)

6. 8 = 43.98 N, C = 72.79 N

Polar Coordinates:

1a. (-2.5, -4.3)

b. (8.4, -9.96)

c. (-19.9, 42.6)

d. (17.2, 22.1)

e. (-15.3, 7.5)

2. r = 5.98, x = 4.44

3a. Point A: (9.4, 58°), Point 8: (4.5, 243°)

b. 13.9 units

4. r = 8.7, y = -6.7

5. X= -21.9, 9 = 215° CC

6. y = -10.3, 9 = 309° cc