Problem 1. The 1-D velocity of a particle is graphed in the figure below as a func- tion of time. Answer each of these questions. Explain carefully and completely your answer for each part. Important: You are obligated to explain how and when you use detailed information from the graph. For example, if you you use the slope of the graph at some point, you need to explain which point(s) you are measuring the slope at and you need to show your calculations for determining the value of the slope. tv (m/s) 10- 9- 8- LUNLOANWAMON 2 3 1 (s ) Important: You must not assume any specific analytical functional form for this graph. Maybe this curve sort of looks like a parabola perhaps. Maybe not. But you cannot rely on this being true to solve this problem. This means that you must use graphical - and not analytical - methods to solve this problem.Problem 1 continues.... Please answer all of the following questions regarding the graph shown on the previous page. Important! You must explain your answers with a sentence or two telling the grader how and why you got the answer you have. Also, if you use information from the graph to answer the question, you must show a sketch or a marked copy of the graph that indicates how you used the graph. For example, if you calculate a slope, you need to show us a sketch that indi- cates how you measured (ideally with a ruler) the quantities that go into your calculation for the slope. . Part (a): Three students make statements: Student 1: "At time t = 0 we know the particle has zero velocity because particles must start at rest." Student 2: "No, you are wrong. In fact at time t = 0 we know the particle is already moving in the negative direction. In fact the particle is slowing down." Student 3: "No, you are both wrong. At time t = 0 the particle is speeding up. We know this because the slope is positive." Only one of these three students is correct. Which student is correct and why? Explain your answer. . Part (b): Is the acceleration of the particle constant during this the entire interval shown here? How do you know this? Explain your answer. . Part (c): Consider the following equation: (t) = rot vot+ zat2 In principle, can we apply this equation to describe the motion of the particle during this entire interval? Yes or No? How do you know this? Explain your answer.- Part (d): Are there one or more instants in time that corresponds to zero ac- celeration of the particle? If so. indicate the tirne(s) [in seconds) and explain: How do you know that the indicates timets) corresponds to zero acceleration? Very important: Explain your answer. . Part (e): Is there any single instant in time that corresponds to the minimum position of the particle? If so. what time is this and how do you know that this is the minimum position? Explain your answer. - Part {1'}: Estimate the acceleration of the particle at the instant corresponding to t : ll] seconds. Important: You must explicitly show your numerical calculations in a complete way and you must explicitly reference the graph to explain how in terms of some graphical method you used the graph to calculate this. Be careful with the units! - Part (g): Harder and more time consuming: Estimate the nal position of the particle :r.r{t} at time t. : 3.6 seconds relative to the initial position :1: 2 f} at t v 0 seconds. There are seyeral approximately correct graphical ways to do this. As long as you are consistent. any method that giyes reasonable accurate results is acceptable. Again. you must use words and use the graph to explain how in terms of some graphical method you you calculate this. Clearly demonstrate to the grader how you obtained your answer. Be careful with the units! (Homework assignment continues next page...) Problem 2. A heavy package of given mass M is released at rest to fall straight down from a hovering helicopter located at a given height H meters from the ground. . Part (a): - How much time t will pass from when the package is released to when it hits the ground? Important: use the equations of Free-Fall Kine- matics to solve this problem symbolically in terms of the given parameters. Here the given parameters are M and H. You can also always assume that any physical constants such as g are also given). Assume you an ignore air resistant. Explain what you are doing. Show your work. Put a box around your final symbolic answer. . Part (b): - How fast will it be moving at the instant just before it hits? Again, use the equations of Free-Fall Kinematics to solve this problem symbolically in terms of the given parameters. Explain your work. Put a box around your final symbolic answer. . Parts (c): - Now suppose you are told that the mass of the heavy box is specified as M=47.84 kilograms and the release height of the helicopter is given as H = 52.8 meters. Plug these numbers into the symbolic answer you just obtained above to calculate the speed of the box numerically. Be sure to take care with significant digits and physical units. Please put a box around your numerical answer. Important! The method outlined here is how we want you to solve every "story problem" in this course. Specifically: . Use symbolic variables (such as M and h to represent given parameters. Al- ways solve in terms of symbolic variables. Do not plug in numbers. Do this even if you are only given numerical values. . Solve the problem in terms of given symbolic parameters and any known physical constants, such as g. Your answer should be an equation where time is given in terms of the given symbolic parameters only: t = f(M, h, g). Put a box around your symbolic solution. Note that it is possible that the final solution will not explicitly depend on all of the given parameters. (Problem 2 continues next page...)(Problem 2 continues:) . Once you have solved the problem symbolically, then and only then should you go ahead and plug in numerical values for the given parameters. Be sure to include appropriate units, and present your answer in terms of an appro- priate number of significant digits. Put a box around your final numerical solution (with units). Very important Reminder: Put a box around your symbolic solutions. Put a box around your final nu- merical solutions (with appropriate SI units). You must always put a box around both your final sym- bolic and/or numeric solutions. (Homework assignment continues next page...)Problem 3. Jose Ramirez swings the bat. but hits a popup. The baseball has a vertical upward velocity of H; 23.? meters per second when the ball leaves his bat. Relative to his bat. how high in the air will the baseball travel? How much total time will elapse before the ball is caught by the opposing catcher?l Assume the ball is caught at the same vertical location that it was hit. Ignore air resistance. - As always. rst solve the problem symbolically. You want expressions for both the height and the time in terms of the given parameters (here just cu} and and relevant physical constants (here just 54). Put a around your symbolic solution. - Now that you have solved the problem symbolically. their and only rhea should you go ahead and plug in numerical values (with appropriate units) for the given parameters and constants. Put a around your numerical solution. As always. explain your work. Very important Reminder: Put a around your symbolic solutions. Put a around your nal nu- merical solutions (with appropriate SI units). You must always put a box around both your nal sym- bolic and/or numeric solutions. {Homework assignment continues next page...) Problem 4. The dwarf planet Sedna is at a distance D : 8-5 AU (Astronomical Units} from the Earth. You want to y a spaceship to Sedna. Suppose you have a spaceship that is able to linearly accelerate in either a positive or a negative direction at exactly A r [1.238 misi. 1l'ou plan a trip where you move with constant positive acceleration in a straight line toward Sedna. speeding up the whole time. and then. at precisely the halfway point. you turn the ship around and accelerate in the opposite direction e. 3.. with negative neeeierrition of the same magnitude so that you are now slowing down. By symmetry. if you reverse acceleration at the halfway point. you should arrive at your destination with zero velocity (right?). Note: in this problem we are completely ignoring the practical reality of how one might build a spaceship that can sustain constant acceleration for an arbitrary period of time. We are also ignoring the motion of the Earth and of Sedna during the trip. - Part (a): How long will the trip take? Important: You must given a syru- bolic answer in terms of the given parameters (here a. and D corresponding to the acceleration of the spaceship and the distance traveled to Sedna. your symbolic answer. Also give a numerical answer in terms of seconds. Be sure that the answer has an appropriate number of signicant digits. your numerical answer. Also convert your answer in seconds to an answer in units that are most appropriate to actual length of the trip (days. years. etc.) Again box your answer. Show your work. Hint: you will need to convert from \"AU" to meters. - Part {b}: What is the maximum velocity of the spaceship during the trip'? Again. present your answer symbolically in terms of given parameters. this. and then plug in to get a numerical answer in terms of appropriate SI units. and then this. Be sure to explain your work. - Part {c}: Draw a qualitative but clear sketch showing plots of the accel eration. the velocity and the position of the spacecraft as a function of time during the entire trip from Earth To Sedna. For each plot. explain in a sen tence or two how you determined the shape of the plot. Problem 5. An apple sits at rest on top of a book which in turn sits on a table that sits on the oor. The mass of the apple is given as my = T6 grams and the mass of the book is mg : 1.0? kg and the mass of the table is my- 13.4 kg. - Part {a}: Determine the net force on the book. Hint. This is a conceptual question. Literally: no calculations are required. whatsoever. Just apply Newton's Second Law. - Part {b}: Draw complete and proper Free Body Diagrams for both the apple and the book. Include only forces that are listed on the \"Force Menu" (see Course Document #04. page 7'} Each and every force on each object must be consistently and properly labeled. For example. if you want to indicate the force of 1Weight on the apple then you should label such a force as ill-'24 which means \"the Weight force on Body A.\" Likewise if you want to label the Normal force on the book due to the apple you would label this #34. - Part {c}: Apply Newton's Second Lavtr together with your two Free-Body Diagrams to calculate the magnitude of each and every individual force on each of the two object [apple and book}. Ignore air resistance and friction. Explain your work. Note that you must invoke and then apply Newton's Sec ond Law to receive proper credit. Note: you must rst set up and solve this problem symbolically, not numerically. In other words, give your answers in terms of the given parameters my. T313 and perhaps \"1]", and the physical constant y. Only after you have done this, should you plug in the numbers to your answer in terms numbers with appropriate units at the very end. Aa- swers that are solved numerically and not symbolically will not receive full credit. {Problem 5 continues next page...) (Problem 5 continues:) . Parts (d), (e) and (f)): - Repeat parts (a), (b) and (c) for Problem 5 above ex- cept now assume that table is sitting in an elevator that is accelerating upward with a given acceleration of precisely An = 6.96 m/s'. Be sure to explain your work. Note that you must explicitly invoke and then apply Newton's Second Law to receive proper credit. Note that students are specifically disal- lowed from invoking "fictitious" forces of any kind. Students are disallowed from "changing the value of g" or anything like this. Students must solve the problem using Newton's Second Law and they must solve the problem us- ing kinematic variables as defined in an inertial reference frame only. Note: again you must set up and solve this problem symbolically, first, not numer- ically. Plug in numbers only at the very end.Problem 6. A Kinematics Question that looks like a Dynamics Question A man swings a bucket of water around in a vertical circle using his arm so that the bucket moves with constant speed in a perfect circle. The radius from the shoulder to the bottom of the bucket is Rh. The mass of the bucket is r-U. The mass of some water in the bucket is raw. The bucket moves with speed Vb. Consider three cases [a] when the bucket is at the bottom of the arc (the man's arm points down relative to his shoulderL [b] the bucket is at the top of the arc [the manis arm points up and the bucket is upsidedown) and {c} the bucket is halfway between [the mans arm is straight out horizontal}. For each of these three cases determine the magnitude and direction of the acceleration of the water in the bucket. When we say \"magnitude" we want an expression in terms of the given parameters. When we say \"direction" we mean the direction of the {vector} 5'1 1.: accelerationi corresponding to \"up , down'Z \"left" and \"right" for example. l-Iint: Read the problem carefully. Consider: Is this a kinematics question or a dynamics