AP PHYSICS SUMMER ASSIGNMENT There are two parts of the summer assignment, both parts mirror the course. The first part is problem solving, where there are 14 math problems that you are given to solve which you should start at the beginning of the summer break and work on throughout the summer. The second part entails reading Chapter 2 of the textbook and answering a series of Reading Questions, which you should not start until the last two weeks of the summer break. PART I (START AT THE BEGINNING OF THE SUMMER BREAK) Solve the following 14 problems Do not try to solve the problems on this paper- use separate pieces of paper and get used to using separate pieces of paper. Think about these problems, they are each designed to make you think about mathematics. If you find yourself looking up solutions on the internet, you have missed the point, which is to learn through struggling. This set of problems requires only algebra and geometry and is meant to prepare you for the course because the essence of the course is in the art of problem solving. Do not put this off to the last week of the summer. Start working on these problems the first week of July. Don't be surprised or discouraged if you cannot solve a problem, these are hard problems. Find a partner and pick a day in the week to work together and complete these problems with them If you have questions or get stuck don t be afraid to email me: zaid.khalil@pequannock.org In Physics, you will be challenged daily in the art of problem-solving. Part of the art of problem solving is seeing deeper connections between different parts of mathematics, for instance the connection between algebra and geometry. Another aspect of the art of problem solving is proving certain relationships. For instance: Geometrically the product of two numbers a and b is represented by the area of a rectangle of sides a and b. b ab a Furthermore the distributive property says the following that the product of a and (b+c) is represented by the area of the following c ac b ab a 1) Geometrically show both cases of the perfect squares identity: ( x y) x 2xy y 2 2 2 2) Geometrically show the difference of squares identity: ( x y)( x y) x y 2 2
3) Use principles of similarity on the right triangles below to show that : x y r 2 2 2 r b α a x c 4) Algebraically show that for any two numbers x and y that: (x+y)/2> (xy) (1/2) hint: The square of any quantity must be greater than or equal to zero. Consider the quantity (x-y) 2 y 5) Suppose AB is a diameter of a circle and C is a point on the circle different from A and B as in the picture below: Show that the angle ACB is a right triangle. hint: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. 6) Show geometrically that for any two numbers x and y that: (x+y)/2> (xy) (1/2) using this figure: x y hint: Use the result of the prior problem to begin this proof.
7) Show that every quadratic equation 2 f () x ax bx c can be reduced to a difference of squares: 2 2 2 2 b b 4ac f ( x) ax bx c a x hint: Use completing the square 2 2a 4a 8) Find the Sum: 1 1 1 1 1 1.... +.. 1 2 2 3 3 4 98 99 99 100 n 2+ n 1 + 1 n 1+ n Hint: Consider multiplying each term by a strategically chosen 1, i.e. rationalizing denominators. 9) Show that the sum of 1+2+3+ +n-1+n = n(n+1)/2. Hint: Consider arranging the sequence 1,2,3,,n, replicate it and rearrange it underneath the original sequence and add the two. 10) N cubes with edge lengths,,,, n-1, n are stacked as shown. 1 2.... There are n cubes are stacked one on top of another whose length is 1, 2, 3,., n-1, n respectively n-1 n Find the length of the portion of contained in the cube with edge n? hint: Use principles of similarity and the results from the previous problem.
11) The definition of a parabola is: the set of points on a curve that are equidistant from a point interior to the curve (focus) as they are to a line exterior to it (directrix). Show that this definition is equivalent to y=ax 2 +bx+c, and find the values of a,b,&c in terms of the vertex of the parabola; i.e. (h,k). 12) The definition of an ellipse is: the set of all points on a curve surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. Show that this definition is equivalent to x 2 /a 2 + y 2 /b 2 =1 where 2a is the length of the major axis and 2b is the length of the minor axis and the origin is the intersection of the two axis. 13) Estimating π: A circle is embedded between two regular hexagons as shown below, use the image to give an upper and lower estimate π: 14) A regular hexagon has side length 2r. Congruent arcs with radius r are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. What is the ratio of the area of the shaded region to the area of the unshaded region?
AP PHYSICS SUMMER ASSIGNMENT PART II (DO NOT START UNTIL LAST TWO WEEKS OF SUMMER BREAK) Learning to read the textbook to extract information and understand theory is an important component of this course. The textbook and your teacher are the primary resources in this class. This is why it is imperative to do the reading and complete the associated reading questions that accompany every chapter. To keep you on task with taking the reading seriously, you will be given periodic reading quizzes on the textbook. The first component of the summer assignment entails Reading all of Chapter 2 in the Giancoli Textbook, and answering the following 25 conceptual reading questions that go along with it. I have included the answers to the odd numbered reading questions at the end of this document. For example the first question is as follows: 1) Which statement is correct about the relationship between the average speed and the magnitude of the average velocity for any motion? A) The average speed is always equal to the magnitude of the average velocity. B) The average speed is always one-half the magnitude of the average velocity. C) The average speed is always less than or equal to the magnitude of the average velocity. D) The average speed can be less than, greater than or equal to the magnitude of the average velocity. E) The average speed is always greater than or equal to the magnitude of the average velocity. Difficulty: 1 Section Reference: Sec. 2-2 Included at the bottom is the difficulty of the question and the section in the textbook which you should reference if you do not know the answer. For the odd numbered exercises try to answer the question with a good effort before you check the answer at the end. This will help you have greater confidence in your choices for the even numbered questions. When you return from the summer break you will have a 15 Question Reading Quiz for Chapter 2, so be prepared and take this seriously. READING QUESTIONS 1) Which statement is correct about the relationship between the average speed and the magnitude of the average velocity for any motion? A) The average speed is always equal to the magnitude of the average velocity. B) The average speed is always one-half the magnitude of the average velocity. C) The average speed is always less than or equal to the magnitude of the average velocity. D) The average speed can be less than, greater than or equal to the magnitude of the average velocity. E) The average speed is always greater than or equal to the magnitude of the average velocity. Difficulty: 1 Section Reference: Sec. 2-2 2) Which statement is correct about the relationship between the instantaneous speed and the magnitude of the instantaneous velocity? A) The average speed can be less than, greater than or equal to the magnitude of the average velocity. B) The average speed is always one-half the magnitude of the average velocity. C) The instantaneous speed is always equal to the magnitude of the instantaneous velocity. D) The instantaneous speed is always greater than or equal to the magnitude of the instantaneous velocity. E) The average speed is always less than or equal to the magnitude of the average velocity. Difficulty: 1 Section Reference: Sec. 2-3
3) The slope of a tangent line at a given time value on a position versus time graph gives A) instantaneous velocity. B) instantaneous acceleration. C) displacement. D) average velocity. E) average acceleration Difficulty: 1 Section Reference: Sec. 2-3 4) If the position versus time graph of an object is a vertical line, the object is A) moving with constant non-zero acceleration. B) moving with constant non-zero speed. C) moving with infinite speed. D) at rest. E) none of the above Difficulty: 1 Section Reference: Sec. 2-3 5) Suppose that an object is moving with constant acceleration. Which of the following is an accurate statement concerning its motion? A) In equal times it moves equal distances. B) In equal times its speed changes by equal amounts. C) In equal times its velocity changes by equal amounts. D) The object is not moving; it is at rest. E) A statement cannot be made without additional information. 6) At a given instant, the acceleration of a certain particle is zero. This means that A) the velocity is decreasing. B) the velocity is zero. C) the velocity is constant. D) the velocity is not changing at that instant. E) the velocity is increasing. 7) Suppose that a car traveling to the East (+x direction) begins to slow down as it approaches a traffic light. Make a statement concerning its acceleration. A) The car is accelerating, and its acceleration is negative. B) The acceleration is zero. C) The car is accelerating, and its acceleration is positive. D) The car is decelerating, and its acceleration is positive. E) The car is decelerating, and its acceleration is negative. 8) A car is traveling north at 20.0 m/s at time t = 0.00 s. The same car is traveling north at 24.0 m/s at time t = 8.00 s. What statement is necessarily true about the acceleration of the car? A) The car undergoes constant acceleration of 0.500 m/s2 during the time from t = 0.00 s to B) The average acceleration of the car is 4.00 m/s2 during the time from t = 0.00 s to C) The average acceleration of the car is 0.500 m/s2 during the time from t = 0.00 s to D) The car undergoes constant acceleration of 4.00 m/s2 during the time from t = 0.00 s to E) The car has zero acceleration during the time from t = 0.00 s to
9) The slope of a line connecting two points on a velocity versus time graph gives A) instantaneous acceleration. B) displacement. C) instantaneous velocity. D) average acceleration. E) average velocity. 10) The slope of a tangent line at a given time value on a velocity versus time graph gives A) average velocity. B) instantaneous acceleration. C) average acceleration. D) displacement. E) instantaneous velocity. 11) If the velocity versus time graph of an object is a straight line making an angle of 30 degrees with the time axis, the object is A) moving with infinite speed. B) at rest. C) moving with constant non-zero acceleration. D) moving with constant non-zero speed. E) none of the above FIGURE 2-2 12) The motion of a particle is described in the velocity vs. time graph shown in Fig. 2-2. We can say that its speed A) remains constant. B) increases and then decreases. C) increases. D) decreases. E) decreases and then increases.
FIGURE 2-3 13) Fig. 2-3 shows the velocity of an object as a function of time. Which graph best represents the acceleration as a function of time? A) B) C) D) E) none of the above 14) Under what condition is average velocity equal to the average of the object's initial and final velocity? A) This is impossible. B) The acceleration must be constantly increasing. C) The acceleration must be constant. D) This can only occur if there is no acceleration. E) The acceleration must be constantly decreasing. Difficulty: 1 Section Reference: Sec. 2-5
15) When is the average acceleration of an object equal to the instantaneous acceleration? A) only when the acceleration is increasing at a constant rate B) only when the acceleration is decreasing at a constant rate C) never D) always E) only when the acceleration is constant Difficulty: 1 Section Reference: Sec. 2-5 16) An object is moving with constant non-zero velocity on the +x axis. The position versus time graph of this object is A) a parabolic curve. B) a horizontal straight line. C) a vertical straight line. D) a hyperbolic curve. E) a straight line making an angle with the time axis. Difficulty: 1 Section Reference: Sec. 2-5 17) An object is moving with constant non-zero acceleration on the +x axis. The position versus time graph of this object is A) a straight line making an angle with the time axis. B) a vertical straight line. C) a hyperbolic curve. D) a parabolic curve. E) a horizontal straight line. Difficulty: 1 Section Reference: Sec. 2-5 18) An object is moving with constant non-zero acceleration on the +x axis. The velocity versus time graph of this object is A) a parabolic curve. B) a vertical straight line. C) a straight line making an angle with the time axis. D) a hyperbolic curve. E) a horizontal straight line. Difficulty: 1 Section Reference: Sec. 2-5
FIGURE 2-5 19) A plot of position as a function of time is shown in Fig. 2-5. Which graph represents the acceleration as a function of time? A) B) C) D) E) Difficulty: 1 Section Reference: Sec. 2-6 20) A stone is thrown straight up. When it reaches its highest point, A) its velocity is zero and its acceleration is not zero. B) neither velocity nor acceleration can be determined without additional information. C) both its velocity and its acceleration are zero. D) neither its velocity nor its acceleration is zero. E) its velocity is not zero and its acceleration is zero. Difficulty: 1 Section Reference: Sec. 2-7
21) Which graph below could represent the motion of the object described in the following sentences? The object that starts its motion with a constant velocity of 2.0 m/s east. After 3.0 s, the object stops for 1.0 s. The object then moves toward the west a distance of 2.0 m in 3.0 s. The object continues traveling in the same direction, but increases its speed by 1.0 m/s for the next 2.0 s. A) B) C) D) E) None of the above graphs could represent the motion described. Difficulty: 3 Section Reference: Sec. 2-6 22) The area under a curve in a velocity versus time graph gives A) displacement. B) distance traveled. C) speed. D) velocity. E) acceleration. Difficulty: 2 Section Reference: Sec. 2-9 23) A ball is thrown straight up, reaches a maximum height, then falls to its initial height. Make a statement about the direction of the velocity and acceleration as the ball is coming down. A) Both its velocity and its acceleration point upward. B) Both its velocity and its acceleration point downward. C) Its velocity points upward and its acceleration points downward. D) Neither velocity nor acceleration can be determined without additional information. E) Its velocity points downward and its acceleration points upward. Difficulty: 1 Section Reference: Sec. 2-7
24) Two objects are dropped from a bridge, an interval of 1.0 s apart. As time progresses, the difference in their speeds A) increases at first, but then stays constant. B) decreases at first, but then stays constant. C) increases. D) remains constant. E) decreases. Difficulty: 3 Section Reference: Sec. 2-7 25) Which of the following graphs could possibly represent the motion as a function of time of an object in free fall? A) B) C) D) E) Difficulty: 1 Section Reference: Sec. 2-7
Answers to Odd Reading Questions 01) E Difficulty: 1 Section Reference: Sec. 2-2 02) Difficulty: 1 Section Reference: Sec. 2-3 03) A Difficulty: 1 Section Reference: Sec. 2-3 04) Difficulty: 1 Section Reference: Sec. 2-3 05) C 06) 07) E 08) 09) D 10) 11) C 12) 13) D 14) Difficulty: 1 Section Reference: Sec. 2-5 15) E Difficulty: 1 Section Reference: Sec. 2-5 16) Difficulty: 1 Section Reference: Sec. 2-5 17) D Difficulty: 1 Section Reference: Sec. 2-5 18) Difficulty: 1 Section Reference: Sec. 2-5 19) C Difficulty: 1 Section Reference: Sec. 2-6 20) Difficulty: 1 Section Reference: Sec. 2-7 21) D Difficulty: 3 Section Reference: Sec. 2-6 22) Difficulty: 2 Section Reference: Sec. 2-9 23) B Difficulty: 1 Section Reference: Sec. 2-7 24) Difficulty: 3 Section Reference: Sec. 2-7 25) D Difficulty: 1 Section Reference: Sec. 2-7