ࡱ> JLIknopqrstuv _ zbjbj *9b9b244444HHH8ldH)Ph~~~YYY((((((($*Z-(4YYYYY(44~~f)###Y4~4~(#Y(##R'O(~RY4({(|)0)1(--<O(-4O(,YY#YYYYY((#YYY)YYYY-YYYYYYYYYB : IB Physics Math Assessment with Answers The purpose of the following 10 questions is to assess some math skills that you will need in IB Physics. These questions will help you identify some math areas that you may want to review as you take IB Physics. As always, select the best answer from the choices given. Consider the following equation:  EMBED Equation.3  where a > 0, b > 0, and d > 0. We are looking for a solution for x where 0 < x < d. It is impossible to solve this equation except with numerical techniques because it involves the reciprocals of polynomials and there is no general formula for that. There is exactly one solution and it can be found using a linear equation. There are exactly two solutions that are found using the quadratic formula. It is possible to write an expression for the solution(s) using the quadratic formula, but it is impossible to know which one(s) (if any) satisfy the condition on x without knowing more about a, b, and/or d. None of the above. ANSWER The correct answer is B. Most people taking this test realize that the equation can be put into the following form:  EMBED Equation.3  There is a great temptation at this point to put this equation into standard quadratic form and use the quadratic formula. The problem can be solved this way, but it is easier simply to take the square root of both sides, which gives  EMBED Equation.3  Since the conditions on the problem guarantee that both sides of this equation are positive, we dont worry about the negative square roots. The solution is  EMBED Equation.3  Could we solve this using the quadratic formula? Yes, but neither C nor D is correct. After some manipulation, the solutions using the quadratic formula are  EMBED Equation.3  There are two solutions to this equation. One is the correct solution given above. The other is an incorrect solution:  EMBED Equation.3  If a < b, then x > d. If a = b, then x is undefined. If a > b, then x <0. In any case, this solution does not satisfy the conditions on x. Therefore, there are not two solutions (choice C is incorrect) and it is possible to select the correct solution without knowing more about a and b (choice D is incorrect). RELEVANCE TO IB PHYSICS We will use this equation or similar equations to find points in space where two gravitational or electric forces cancel each other. In questions 2-4, the function f(x) is continuous and differentiable on the interval [a,b]. That means that the graph of f(x) is a continuous curve with no breaks and the slope of f(x) is well-defined. When we speak of the slope of f(x), we mean the slope of the graph of f(x) versus x. If the slope of f(x) is negative in [a,b], then  EMBED Equation.3  at all points in [a,b].  EMBED Equation.3  at some point in [a,b].  EMBED Equation.3 . All of the above. None of the above. ANSWER C is the correct answer. If the slope of a function is negative throughout an interval, it is decreasing. A negative slope does not imply anything about whether the function itself is positive or negative. RELEVANCE TO IB PHYSICS We study graphs of displacement and velocity in IB Physics and draw conclusions from them. If f(x) < 0 in [a,b], then the slope of f(x) is always positive at all points in (a,b). is always negative at all points in (a,b). must be zero at some point in [a,b]. All of the above. None of the above. ANSWER E is the correct answer. The fact that a function is negative does not imply anything about its slope. RELEVANCE TO IB PHYSICS We study graphs of displacement and velocity in IB Physics and draw conclusions from them. In the interval [a,b], the slope of f(x) is constant. The average value of f(x) in that interval is  EMBED Equation.3 .  EMBED Equation.3 . the area under the graph of f(x) between a and b, divided by (ba). All of the above. None of the above. ANSWER D is the correct answer. C is the definition of the average of a function. A and B are also true for the special case of a function that has a constant slope, or in other words the graph is a straight line. RELEVANCE TO IB PHYSICS We can solve many kinematics problems if we plot velocity versus time and take the area under the curve as displacement. These math facts are particularly useful for motion with constant acceleration. In the graph below (not necessarily to scale), f(x) is defined in the interval [a,b] by the straight line segments connecting the points as shown.  EMBED Zoner.Draw.Document  The area under the graph is always positive or zero. is zero if p = q. can be calculated using area formulas for triangles and rectangles. All of the above. None of the above. ANSWER C is the correct answer. A is not true because the area under the curve of a negative function is negative. B is not true because we dont have enough information about the locations of the points along the horizontal axis to draw that conclusion. While in general we need calculus to compute the area under a curve, in the special case of curves that can be broken into line segments we can use simple geometric area formulas instead. RELEVANCE TO IB PHYSICS We use the concept of area under a curve in IB Physics to solve problems involving displacement, energy, and impulse. The vector  EMBED Equation.3  shown below has length c. Its angle measured counter-clockwise from horizontal axis is q and its angle measured clockwise from the vertical axis is f.  EMBED Zoner.Draw.Document  The vertical component of  EMBED Equation.3  is given by c sin(q). c cos(90 q). c cos(f). All of the above. None of the above. ANSWER D is the correct answer. The equivalence of A, B, and C is a basic fact of trigonometry. RELEVANCE TO IB PHYSICS All IB Physics students should be comfortable with basic trigonometry for finding components of vectors. In the diagram below, the vectors  EMBED Equation.3  and  EMBED Equation.3  are at right angles to each other. The length of  EMBED Equation.3  is c and the length of  EMBED Equation.3  is d. The horizontal and vertical components of  EMBED Equation.3  are a and b respectively.  EMBED Zoner.Draw.Document  The vertical component of  EMBED Equation.3 , e, is a d / c. b d / c. b d / a.  EMBED Equation.3 . None of the above. ANSWER A is the correct answer. In order to solve this problem, it helps to add two named angles to the diagram:  EMBED Zoner.Draw.Document  We are looking for e, so we need sin of f. We note that q is the complementary angle of f, so sin(f) = cos(q) = a/c . The final answer is d sin(f) = a d / c. RELEVANCE TO IB PHYSICS This question requires math skills you will need to solve inclined plane problems and to calculate vector directions for gravitational and electric forces. Consider two vectors,  EMBED Equation.3  and  EMBED Equation.3 , and their sum,  EMBED Equation.3  =  EMBED Equation.3  +  EMBED Equation.3 . Which orientation of the vectors shown below gives the smallest magnitude (length) of  EMBED Equation.3 ? (The magnitudes of  EMBED Equation.3  and  EMBED Equation.3  dont change in the diagrams, but dont assume that they are drawn to scale.)  EMBED Zoner.Draw.Document   EMBED Zoner.Draw.Document   EMBED Zoner.Draw.Document  It doesnt make any difference since the magnitudes of  EMBED Equation.3  and  EMBED Equation.3  dont change. There is no way to tell without knowing the magnitudes of  EMBED Equation.3  and  EMBED Equation.3 . ANSWER A is the correct answer. RELEVANCE TO IB PHYSICS We use vectors a lot in IB Physics. This question is trying to get you to think about how vectors are aligned to give the minimum sum: opposite directions. The alignment of vectors is critical when we consider speeding up / slowing down, centripetal acceleration, torque, work, and many other topics. (Challenge problem.) The vectors  EMBED Equation.3  and  EMBED Equation.3  are given as shown below, with lengths c and d respectively. The angle of  EMBED Equation.3  with respect to the reference line is q and the angle of  EMBED Equation.3  with respect to the reference line is f, going counter-clockwise.  EMBED PBrush  The magnitude (length) of the sum,  EMBED Equation.3  =  EMBED Equation.3  +  EMBED Equation.3 , is  EMBED Equation.3 .  EMBED Equation.3 .  EMBED Equation.3 .  EMBED Equation.3 . The magnitude of  EMBED Equation.3  cannot be determined unless the direction of the reference line is known. ANSWER D is the correct answer. The easiest way to get this result is to realize that the magnitude of a vector sum is independent of the choice of coordinate systems. In that case, pick the X axis to align with the u vector. The angle of the v vector with respect to the X axis is (f-q). We take the X and Y components of v using trigonometry, add the X component of u, and finally use the Pythagorean Theorem to get the length of the sum. Just for laughs, we put cos(q-f) instead of cos(f-q) because cos( a) = cos(+a). Note that (q-f) and (f-q) are independent of the angle of the reference line. While we re on the subject, let s go over how to convert vectors from (Length,Angle) coordinates to (X,Y) coordinates and back. First, assume that the angle has been measured in the counter-clockwise direction from the +X axis. This is the standard you will see most often and the one we will use in IB Physics. To convert from (Length,Angle) to (X,Y), we use the following formulas (Let length = L and angle = a.):  EMBED Equation.3   EMBED Equation.3  Converting from (X,Y) to (Length,Angle) is a little trickier. We have to be careful to get the angle in the proper quadrant. First, the easy part. The length is given by the Pythagorean Theorem:  EMBED Equation.3  Calculation of the angle depends on the quadrant we are in. First, the special cases. If X = 0 and Y = 0, then by convention we say L = 0 and a = 0. If X = 0 and Y > 0, then a = p/2 in radians or 90. If X = 0 and Y < 0, then a =  p/2 in radians or  90. If Y = 0 and X > 0, then a = 0. If Y = 0 and X < 0, then a = p in radians or 180 In cases where X > 0, we are in quadrants I or IV. Use this formula:  EMBED Equation.3  In cases where X < 0 and Y > 0, we are in quadrant II. Use this formula:  EMBED Equation.3  (radians) or +180 In cases where X < 0 and Y < 0, we are in quadrant III. 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On computers, you will find these formulas programmed into one function: atan2(Y,X). RELEVANCE TO IB PHYSICS This question is more complex than most vector calculations we will need in IB Physics, but it brings out some important concepts that you will use in later courses at RPI. In any case, make sure you understand how to convert a vector from (Length,Angle) form to (X,Y) form and back. Dick and Jane went into business selling mud pies. They set up two mud pie stands at opposite ends of the neighborhood to maximize their potential customers. On Monday, they sold a total of 100 pies. On Tuesday, Dick took the day off because his sales were so good on Monday. Jane figured that she could triple her sales from Monday if she copied Dick s slick sales techniques. Unfortunately, Dick s customers from Monday all returned their pies to Jane on Tuesday. Jane met her Tuesday sales target, but she only sold 20 new pies after reselling all of Dick s returned pies from Monday. What equations would you use to determine how many pies Jane sold on Monday? J + D = 100 3J = 120 J + D = 100 3J  D = 20 J + D = 100 3J + D = 120 There is not enough information to determine a unique answer. There is contradictory information so there is no answer. ANSWER B is the correct answer. The second equation could also have been written as 3J = D+20. RELEVANCE TO IB PHYSICS We often get pairs of linear equations in IB Physics when we solve problems using Newton s Second Law. Many problems involve objects connected by ropes. A variable representing tension in a rope normally appears in two equations, one for each end of the rope. With the right choice of coordinate systems, the tension term will be + in one equation and  in the other. The easiest way to solve a system of equations like that is simply to add the two equations. (Easy = less likely to make an error.) To solve the two equations in B, add them together to get J + 3J + D  D = 100+20 Solve for J, then substitute J back into either original equation to solve for D.      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