1 Welcome to Honors Physics, a most rewarding class here at East Brunswick High School. The first step to success in this class is the accurate completion of the summer assignment. Why a summer assignment, you ask? This can be answered with your first lesson to combat inertia. Two months at home doing no schoolwork does not make for a strong work ethic come September. When you think about all the important discoveries that were made during summer months (for example, Newton inventing Calculus) it should inspire you to make some discoveries yourself. This summer assignment covers some math review and the first half of our first unit. The math review includes some straight-forward algebra that evolves into algebra with physics equations. You are not expected to know or understand the formulas, just how to manipulate them. This packet will be collected the first day of school, but your quiz on this information will be in the first week so it is your responsibility to be familiar with this material. If you find the material in our first unit difficult, you are encouraged to seek out resources online. The summer assignment is not designed for you to do in one sitting. Working through it a little at a time will help to reinforce the material and will not make it seem so overwhelming. Good luck and see you in September. As soon as you are officially registered, a classroom moodle will be available to you if you are in Mrs. Eastep s class. If you have her as a teacher and are still not receiving access by August 20, please either Math Review Within any science, inevitably you will need to solve for specific variables and you will sometimes need to rearrange equations before inputting any numbers. This is definitely a skill that you will need to use this year. Below are some examples that you should be able to complete with very little difficulty. 1. f = e pd, for e 13. F m v net and F net = m a, for v in terms of a, m, and t = t 2. 6 t + 62s = 1 2 (3t 42s), for t 14. U = qv ; KE = 1mv 2 2 ; U = K E, for v in terms of q, V, and m I 1I2 V = 2 V m + n + 2 p = 3, for m 15., for V 4. 3x+y c = 4, for c 16. n i sinθ i = n r sinθ r, for θr 5. 5xy+n 2 = 6, for y f = d 1 o + d 1, for d i i 6. v = 4 3 r, for r 18. m gh = 1mv kx 2 2, for x 7. 3 c + 5 d = 7d 6 c, for d 19. 1mv 2 = m gh, for h 8. 12g + 31h = 8 g, for h 9. v = v o + a t, for vo 10. mv 2 r = m a, for v Gm 1 m 2 m v r = 2 r 11., for r Gm m r = 1 2 m v, for m Vectors vs. Scalars I. Scalars are quantities that are fully described by a magnitude (numerical value) alone. A. Some scalars can be positive or negative B. A negative scalar represents a quantity being taken out of the system. 2
2 C. A positive scalar represents a quantity being added to a system. D. A positive scalar is always greater than a negative scalar. Honors Physics These are examples of scalars that will be addressed this first unit. 1. Time (t, seconds; s) 2. Length (L, meters; m) 3. Distance (d, meters; m) 4. Speed (s, meters per second; m/s) II. Vectors are quantities that are fully described by both a magnitude and a direction. A. Vectors can be positive or negative B. A negative sign on a vector just represent the direction it is going. C. Conventions for Direction: a. Negative directions include Left, Down, South, West. b. Positive directions include: Right, Up, North, East. D. To determine which one is greater, we look at the value/magnitude and not the sign. Some examples of vectors are: 1. Displacement ( Δ x, meters; m) 2. Velocity (v, meters per second; m/s) III. Direction A. Since vectors need a direction to fully be described, we must assign +/- values and create a convention. 1. X axis. Positive vectors point to the right Negative vectors point to the left. 2. Y-Axis Positive vectors point upwards. Negative vectors point downwards. 3. What about when they are in the x and y direction? *If the vector is on a graph, you can read off the graph the x-component of the vector and the y-component of the vector. If it is not on a graph, you need to use trigonometry to calculate the x and y components. This will be addressed later. *When a vector has a combination of both x and y components, the vectors must be separated. IV. Adding Vectors A. Tip to Tail Method Example 1:
3 + = = **After you connect the vectors, the resultant vector starts at the tail of the first and ends at the head of the final vector. Either way you set them up, notice that the resultant vector is the same.** Practice 1: Sketch in the resultant vectors for each of the problems below. + = + = + = + = B. Adding via mathematical components 1. Vectors have numbers that can be found on a graph; they use a coordinate system. The first number is the x-component is the first number listed while the second number is the y-component. Vector 1: (3,2) The head of the vector is three blocks to the right and two blocks up. Both are positive. Vector 2: (-2,0) The head of this vector is two blocks to the left and not blocks up or down. Vector 3: (-3, -3) The head of this vector is three blocks to the left and three blocks down. Example 2: To add the vectors, all you do is add individual axis. V1 + V2 = (3,2) + (-2,0) = (1, 2) Practice 2: Find the coordinates, add the following vectors and draw the resultant vector, mathematically. VII. The Trig Functions A. Simple Vocabulary
4 1. Instead of using a, b, and c, for the sides, we will change our discussion to opposite, adjacent, and hypotenuse 2. Looking at the triangle to the right, c will always be the largest side, the hypotenuse. 3. To determine the opposite and adjacent sides, we must rely on the angle. The opposite and adjacent side will change depending on which angle you are using. B. The Functions Themselves 1. Now that we know vocabulary between angles and sides, we can start with the full trigonometric functions. SOH-CAH-TOA C. Applying the Trig Functions 1. When using the trig functions, you can solve triangles with two unknowns. 2. To determine the missing sides of this right triangle, write down your knowns: θ = 37 o and c = 8 The adjacent side is a and the opposite side is b. Using the sin function, we can identify side b. sin 37 o = b/8 b = 4.8 Using the cosine function, we identify side c. cos 37 o = a/8 a = 6.4 D. Application to real-life 1. Justin, while camping, decides to go on a hike. He travels 7.50km north, 5.90km west and then makes a direct trip home. A) Graph his trip on the graph provided to the right. B) What are the x and y coordinates the final
5 displacement vector? C) What is the magnitude and direction of each of the legs of his trip as well as his final displacement from his origin? (B) ( 5.90, 7.50)km and C) (9.54km, )) 2. Four young toddlers are all pulling on a single stuffed dog. Robert tugs with a force of (3.91, 2.50) N. David pulls with a force of ( 3.65, 1.35)N. Austin pulls with a force of (3.12N, 145 o ). Sammy tugs with a force of (4.23N, 293 o ). A) Convert each of the forces to x,y coordinates. B) Find the resultant force (magnitude and direction) that the dog is being tugged with. (1.15N, 236 o ) Part 1: The Conceptual Basics of Motion I. The Definitions A. Where are you?!?!?! 1. Position is the actual location with respect to a reference point. We will always use the origin of (0,0) when determining position of an object at a particular point in time. It is a vector and therefore to correctly establish an accurate position, you
6 must include direction as well as distance from the origin. Honors Physics a. The sign of position indicates whether the object is to the right (+) or the left (-) of the origin. B. What happens if you move?!?!? 2. Distance is a measure of the physical length between two locations measured along the actual path connecting them. Distance is a scalar quantity and therefore is never negative since it has no direction. A car odometer measures distance, it continues to add distance to the quantity already on its meter to indicate the total distance traveled by the car. Driving in reverse does not subtract distance from the odometer, since direction is not significant when finding the distance an object travels. 3. Displacement is change in position between two locations measured along the shortest path connecting them. x f -x i = Δ x= displacement. Notice, all s are: final initial. Displacement is a vector quantity and is dependent on the direction of travel. When determining the total displacement of a journey that has more than one leg, find the resultant displacement. The resultant displacement is determined by using the initial position and the final position. Displacement does NOT depend on the path! The sign of displacement indicates whether the movement is in a positive + direction (usually we choose to define this as to the right) or the negative (-) direction (usually left) of its original position. Remember, displacement is change in position, x f -x i = Δ x. The signs of these directions are arbitrary and are used in this way the vast majority of the time because it is easier. If it turns out that making left + makes the work easier then we do it that way instead. Example 1: Imagine you leave your physics class and walk 10 meters down the hallway to the right before you realize you forgot your homework. You turn around and walk 10 meters down the hallway to the left when your wonderfully helpful teacher walks out to meet you and hand you your homework. What distance did you walk? What was your displacement? Example 2: Angie is flying a remote controlled helicopter. She guides the helicopter 12.0 m north then turns it around and flies it 18.0 m south. She turns the helicopter around and flies it 15.0 m north and finally, flies it 7.0m south before setting it gently on the ground. (N, 52.0m, +2m) a. What direction is positive? b. What total distance did the chopper fly? What displacement did the helicopter accomplish? Cartesian and polar. Give the answer in both. c. What was the position of the helicopter after (i) 12m north, (ii) 18m south, (iii) 15m north, and (iv) 7m south? Cartesian and polar. Give the answer in both. Example 3: Danielle is two laps away from winning the Indy 500. The race consists of exactly 200 laps around a 2.50 mile oval track. (495mi, 0.00mi) a. Considering just the race and not any hazard or pace laps, what distance did Dani travel? b. What displacement did Dani accomplish?
7 Example 4: Object A moves from (4,0) m to (7, 0) m. Object B moves from (7,0) m to (4, 0) m. Object C moves from (-3,0) m to (2, 0) m. Object D moves from (-3,0) m to (-7, 0) m. Object E moves from (-7,0) m to (-3, 0) m. Object F moves from (4,0) m to (-7, 0) m. Label and plot the initial and final position of each object on the number line below. Sketch each displacement vector. Is there any regularity to whether the object is moving towards or away from the origin? Object A has been done for you. For the remaining vectors, make separate lines for each of them (no lines on top of one another). Fill in the chart summarizing the data from above Object Initial Position (m) Final Position (m) Displacement (m) A B C D E F II. The Difference between speed and velocity 1. Speed is a scalar quantity and will always be positive since it has no direction. It is found by dividing the distance traveled in a given time interval by the length of the time interval. Sometimes a trip has several legs of a journey. In this case you may be asked to find the average speed of the whole trip. This is done by dividing the total distance for the whole trip by
8 the total time for the whole trip. Avg Speed = d tot /t tot Honors Physics 2. Velocity is a vector quantity that is calculated by the change in position (displacement) during a given time interval divided by the time interval. v= Δ x / Δ t. Direction is significant here. The + direction for velocity always matches the + direction for displacement since time is always +. We usually use right and up as +. The signs of these directions are arbitrary and are used in this way the vast majority of the time because it is easier. If it turns out that making left + makes the work easier, then we do it that way instead. Example 5: Grace jogs to her neighbor s house in 28.0 s at a speed of 3.25m/s. Seeing her neighbor was not home, she continues to jog to her boyfriend s house in 36.0 seconds at a speed of 3.00 m/s. What is her average speed for the entire trip? (Remember, average speed is equal to the total distance divided by the total time. You cannot simply average the two speed values.) Practice by Checking your Answers ALWAYS SHOW YOUR WORK 1. You are standing in K-hall waiting for your friend so you may go to physics. You start at a position P that is 6.5m from the classroom door. You walk 1.5m toward the door and then turn and walk 0.8m away from the door. After pausing for 15 seconds, you walk 0.6m toward the door before stopping. (2.9m, -1.3m, 5.2m) a. What distance did you pace while you waited for your friend? b. What displacement did you accomplish while you waited for your friend? c. What was your final position relative to the classroom door after your pacing? 2. Under what conditions is the total distance traveled during a trip equal to the magnitude of the net displacement? 3. Under what conditions is the net displacement equal to zero? 4. When the total distance and the magnitude of the displacement are not equal, is either one always larger than the other? Which one is always larger than the other? Explain why one is always larger or why one is not always larger.
9 5. An object moves at constant speed toward a wall and it travels a distance of 5.00m toward the wall in 3.50s. It reaches the wall and returns at a constant speed that travels a distance of 5.00m away from the wall in 3.50s. (1.43m/s, yes, 0.00m/s, no) a. What is the average speed of the ball? b. Is the speed of the ball as it approaches the same as the speed of the ball as it returns? Why or why not c. What is the average velocity of the ball for the entire trip? d. Is the velocity of the ball as it approaches the same as the velocity of the ball as it returns? Why or why not? 6. Indicate the sign of the position, displacement, and velocity for the following A ball moving to the right at constant speed A block moving to the left while slowing A book moving to the left while speeding up A ball moving to the right while speeding up A ball moving up and slowing A rocket moving up and speeding up Position Displacement Velocity Part 2: Mathematical basics of motion I. Fun with Motion in One Dimension (All motion problems will be dealing with constant speed/velocity) Let s apply our understanding of distance, position, displacement, velocity, speed and acceleration and have some fun with math! 1. First define each term in your own words. What information does each variable tell you? What does it not tell you? Position: Define: What does it tell? What doesn t it tell? Formula? Notes Displacement: Speed: Distance: Velocity: Example 1: At t = 0s an object has a position of -4.0m and without changing direction moves to a position of +6.0m at t = 4.0s. It then moves in a straight line and accomplishes an additional displacement of -8.0m at the t = 8.0s mark. Sketch a
10 number line and diagram the motion for the entire 8 sec period. Then answer the questions. (18m, +2.0m, 2.25m/s 2, +.25m/s, -2.0m) a. What total distance did the object travel? b. What was the object s overall displacement? c. What has been the object s average speed? d. What has been the object s average velocity? e. At t = 8.0s, what is the object s position? Example 2: Jack and Jill run a 100.0m race at constant speed. Jack is 2.00s and 10.0m behind Jill as she crosses the finish line. What is the winner s time for the race? Wow it seems there is not enough information. That happens a lot in this course. Approach the problem from a different angle. (18.0s) Example 3: Two football players, Bobby and Scott, run toward each other from opposite ends of a yard football field. Bobby runs at a constant speed of 8.00 yard/s while Scott runs at a constant speed of 6.00 yard/s. A third player, Tim, starts at the same time as the other two young men, but he begins at the 20.0-yard line at the same end of the field as Bobby. a. If they all arrive at one place on the field at the same time, how much time has passed? (7.14s, 57 yd line, 5.24yd/sec) b. Where on the field do they meet? (Where is your origin/reference?) c. How fast is Tim running? Example 4: Alexis the Astronaut pushes off from a space station, drifting away at 1.17m/s, while the station moves at m/s in the exact opposite direction. Alexis is connected to the space station by a tether cable seconds later, the cable is pulled taut and stops the astronaut. How long is the cable? (60.9m) Practice by Checking your Answers ALWAYS SHOW YOUR WORK
11 1. Bryan sprints for 10.0s at a speed of 8.00m/s, then runs an additional 100.m in 12.0s, and then finally travels an additional 200.m with an average speed of 8.00m/s. What is Bryan s average speed for the entire trip? (8.1 m/s) 2. Hunter gets a new red convertible and decides to drive to her friend s house 250 km away to work on physics homework. For the first half of the distance to her friend s house, she travels at an average speed of 40.0km/h. Hunter wishes to keep the average speed for the whole trip at 50.0km/h. What speed should she average for the second half of the trip? (66.8 km/h) 3.Three bicyclists start a 20.0-km race at 12:00 noon. Gina travels at an average speed of 30.0km/h, Jinnie at 25.0km/h, and Cathie at 22.0km/h. a. At what time does Gina finish the race? (12:40pm) b. How far have Jinnie and Cathie gone when Gina crosses the finish line? (16.7km and 14.7km) 4. Two soccer players run toward each other from opposite ends of a 120.-m soccer field. Trace runs at a constant speed of 10.0m/s while John runs at a constant speed of 7.00m/s. If they both start running at the same time... a. How much time passes before they meet? (7.06s) b. Where on the field do they meet? (70.6m)
12 Part 3: Graphing your Motion I. What is Kinematics? 1. Kinematics is the study of classical mechanics which describes the motion of objects without considering the causes of the motion. To describe motion, kinematics studies the variables such as velocity, distance, time, and acceleration. Ex. v = d/t (a kinematics equation that you have been using) This equation relates velocity, distance and time, but tells us nothing about what, why, how, etc. 2. The motion of objects can be described with mathematical formulas and with graphs of these formulas. Often it is useful to graph the position, velocity, and acceleration of an object versus time to show trends in motion and to supply much information about the motion of an object. 3. We will only consider graphs of objects moving in a straight line in real physical space. II. Position versus Time Graphs Graph 1 Graph 2 Graph 3 Example 1. For ease of discussion each line segment is labeled. The slope of a position time graph is velocity. Slope = y/ x = d/ t = velocity by definition. Any portion of a position vs. time graph with a constant slope indicates the object is traveling at constant velocity. The rate of change of position with respect to time is constant, we call this constant velocity. 1. Does the velocity of object X change from segment A to B? Explain. 2. Does the speed of object X change from segment A to B? Explain. 3. How can you tell if the object has a positive or negative position and a positive or negative displacement? 4. How can you tell if the object is moving in a positive or negative direction? 5. How can you tell if the object has a positive or negative velocity? 6. How can you tell if the object is moving left or right? 7. How can you tell if the object is moving towards or away from an observer at the origin? 8. In which segment(s) is Object X moving away from the origin? 9. In which segment(s) is Object Y moving away from the origin? 10. In which segment(s) is Object Z moving away from the origin?
13 The graphs from the previous page are shown again to the right for you. Please continue to answer the following questions based on the graphs. 11. In which segment(s) is Object Z moving the fastest? How do you know? 12. When is Object Y 6m away from the origin? 13. Which object is at x = 2m first? 14. How far is Object Z from the origin at t = 3s? A B C D E F G 15. Which object takes the least time to reach a position 4m from the origin? 16. Which object is farthest from the origin at t = 2s? 17. Is there an object that eventually returns to the origin and, if so, which one does this and when does it occur? 18. What is the total distance travelled by each of these 3 objects during the full 5-second time interval? 19. What is the displacement of each object between t = 1s and t = 3s? Which object has the largest displacement here? 20. How far (displacement) does X move every second during the last 3 seconds? 21. What direction is X moving during the last 3 seconds? How do you know the direction? 22. Which object moves the greatest distance during a single second? During which 1s time interval does this occur? 23. Which object is moving most slowly between t = 0s and t = 2s? Explain how you know. 24. Which graph has the smallest slope at t = 1s? What does this tell you? 25. What happens to the position-time graph when an object speeds up? 26. What happens to the position-time graph when an object slows down? 27. What happens to the position-time graph when an object reverses direction? 28. What happens to the position-time graph when an object passes through the origin? Answers to the above questions 1. Yes 2. No 3. Location/change in location 4. Slope 5. Slope 6. Need to know conventions 7. Moving toward or away 8. A 9. C and D 10. E 11. G/slope s 13. Z 14. 4m 15. X 16. X 17. Z m, 10m, 9m 19. 0m, 1.25m, 2m 20. 1m 21. Toward the origin 22. Z, X 24. X 25. Larger slope 26. Smaller slope 27. Change in direction of slope 28. Crosses 0 on y-axis
14 Example 2: For each description of a physical situation, identify which of the graphs below could represent the motion of the object. You should specify which direction is positive before choosing a graph. 1. A marble is rolled at a constant speed along a horizontal surface toward the origin. The marble is released at a distance of 1m away from the origin. 2. A block sits at rest on a table 1m above the floor. Take the origin to be the level of the floor. 3. A ball is rolled along a horizontal surface at a constant speed. The ball strikes the wall and rebounds toward the origin. 4. A car is parked on a steep hill. 5. Which represents an object that stops during its motion? How do you know this is the case? 6. Which graph(s) represent(s) objects moving away from the origin? How do you know this is the case? 7. Which graph(s) represent(s) objects moving toward the origin? How do you know this is the case? Answers 1. B 2. D 3. C 4. D 5. C 6. A, C 7. B, C Example 3: Answer the following questions based on position-time graphs. 1. Can the displacement of an object ever be zero? How? How would you know this by looking at a position-time graph? 2. Shown below is the position-time graph of a marble rolling along the floor. It starts at x = 1m at t = 0s and rolls in the positive direction at constant speed. On the same diagram, sketch and label the position versus time graph for each situation described below. a. A marble rolls in the same direction and with the same speed as the original, but starts at the origin instead. b. A marble rolls in the opposite direction with the same speed, starting at x = 0m. c. A marble has the same starting point and direction as the original, but is rolling at twice the speed as the original. d. A marble starts at x = 1m and rolls with the same speed and and direction as the original but doesn t move until t = 4s. e. A marble is stationary at x = 1m. f. A marble starts at the origin, and travels in the same direction as the original but at half its speed. g. A marble starts at x = 1m and travels at half the speed as the original and in the opposite direction. 3. What happens when the position-time graph goes from positive to negative? Does the direction of motion change also?