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- 1. Help Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 1
- 2. Help Since 10 years I am a co-lecturer for mechanics for our Industrial Design students and I have lectured dynamics for our students "Technische Bestuurskunde" for 3 years. I knew their level, in my view, I was going slow through their material, and it worked. So compared to those students the course "Physics" for our math students would be be a piece cake. I was so wrong! Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 2
- 3. Why is Physics difﬁcult for our math students? Because: 1. Their VWO -level for physics is low(er than expected) ? 2. Doing mathematics in a physical environment is something else (it starts with physical insight, it is calculus, and it is about the basics of vectors)? 3. It is a long way from Newton to the solution of a medium physics exercise, because Newton is so general? 4. Interpretations for "common" subjects in physics and mathematics are sometimes different or vague? 5. If you start with a problem "exactness" can be a handicap in some sense? 6. It takes (a lot of) time to ﬁnd a right attitude for understanding the concepts of physics? 7. The time span of a quarter is too short to acquire insight in physics. Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 3
- 4. The course CM and SR: Some characteristics of the course "Mechanics and Special Relativity" for ﬁrst year students mathematics in quarter 3 : ◮ Each week 2 x 2-hours (theory-) lecture and 2 x 2 hour "instruction" for working on exercises, ◮ Teacher for the lectures: Dr. Timon Idema (assistant professor, TU Delft,nano-physics, and founder of a theoretical biophysics group at the Department of Bionanoscience at TU Delft (2012), Instructor: Emiel van Elderen (Diam - Lecturer), ◮ "Mechanics and Special Relativity" is one of the 3 optional subjects offered to our students during the third quarter of the ﬁrst year. The number of participants at the start was ± 28. (The other optional subjects were Electromagnetism, and some course in the ﬁeld of computer science, Algoritm and Data structures ) Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 4
- 5. Contents and Grading: The course consists of 2 parts, Classical Mechanics (CM) and Special Relativity (SR), each 4 weeks. ◮ Book: J.R. Forshaw and A.G. Smith, Dynamics and Relativity. ◮ Topics (CM): The 3 laws of Newton; some Force laws: Hook, Gravitation,...etc.; Free Body diagram; composing (differential) equations of motion for point masses; Rigid (ﬂat) bodies, Conservation Laws for momentum, mechanical energy and angular momentum and some examples of collisions. ◮ Topics (SR): Postulates of Einstein, Time dilatation, Galilei- and Lorentz transformation, four vectors, Minkowski-time space, the famous formula E = mc2 and some attention for collisions (ﬁssion and fusion). ◮ Grading: 5% for answering (feedback fruit electronic -)questions during the lectures, 30% midterm exam (only CM), 65% ﬁnal exam ( CM and SR) Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 5
- 6. My ﬁrst observations 1: ◮ A very good and enthusiastic teacher, some of his quote’s and lecture style: ◮ Like learning math, or playing the piano, learning physics takes practice,... ◮ Now if you do not understand, no worries, thats what we are for... ◮ With feedback fruits the students are very actively involved during the his lectures... ◮ The main reason for the students choice for "Mechanics and Special Relativity" is the special relativity part they want to know ﬁnally what it is about! ◮ During the lectures there were experiments (mostly Classical Mechanical) to illustrate the theory, which the students appreciated very much. ◮ About the instructions, some of the students could make a start with the given exercises, most of them couldn’t, not even after explaining the beginnings. Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 6
- 7. My ﬁrst observations 2: ◮ A third of the students lost their motivation during ﬁrst weeks. ◮ During week 5 (during and because of the midterm exams) the number of attendants of the lectures and instructions dropped to 5, while Timon just made a start with special relativity. ◮ At the beginning of the second part of the course the number of students dropped to 16 to 18 students at most. ◮ Of these students, 14 have got a sufﬁcient grade for the course (ﬁrst time). Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 7
- 8. Example 1: Their VWO -level for physics is lower... Doing mathematics in a physics environment is something else...: . A person throws a ball with a given speed v towards a wall of height h. Which of the following quantities is the maximum distance the person can stand from a wall and still able to throw the ball over the wall? a. gh2 v2 b. v2 g c. v4 g2h d. v2h g e. v2 g 1 − 2gh v2 f. v2 g 1 + 2gh v2 Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 8
- 9. Example 1: a. gh2 v2 b. v2 g c. v4 g2h d. v2h g e. v2 g 1 − 2gh v2 f. v2 g 1 + 2gh v2 ◮ Starting level: They know the basic physical quantities Length, Time .... and their unities. ◮ They could answer "what is the dimension of mvt ?" (one of the ﬁrst questions of the workshop) ◮ They found that all dimensions of the expressions above are Length so a lot of them did not know how to go further? ◮ My tip: make a sketch of the situation and try to make qualitative estimations, it is this crucial aspect of physics which takes a lot of time to learn. ◮ Another tip: what is the maximum distance in your problem without a wall? For lot of students this was a "mission impossible", for explaining this I needed to return to the basics of V.W.O.-physics. Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 9
- 10. Example 2 statics: It takes (a lot of) time to ﬁnd right attitude for understanding the concepts of physics. In general it is a long way from Newton to the solution of a medium physics exercise: Keeping a book up and leaning sticks: Find the extreme values for θ in case of equilibrium! ◮ Starting level: They have seen Newton F = ma and some examples. ◮ Their problem is how to start. "Where" are the forces? (How to Find a Free Body Diagram?) ◮ Another problem: How to handle a parametric solution. What are your parameters and what are your variables? ◮ Recognizing a optimization problem. Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 10
- 11. Example 3 dynamics: It takes (a lot of) time to ﬁnd right attitude for understanding the concepts of physics. In general it is a long way from Newton to the solution of a medium physics exercise Coupled motion, dragging a board, a collision on ice : The same holds for A B Find acceleration! and How much work? Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 11
- 12. Example 4: 2D- collision: Doing mathematics in a physics environment is something else (At a ﬁrst year level) it is calculus and basics of vectors) A 2D-problem: a moving puck and a stick on an ice-ﬂoor D v0 massamiddelpunt stick b massamiddelpunt stick massamiddelpunt stick + puck a) b) What is happening after the collision? The last exercise is a 2D-problem . The student has knowledge of linear algebra and thus about vectors. This was not enough to understand what is happening here. We need dL dt = τ and geometrical interpretations (for the cross product) in this context of L = r × mv and τ = r × F. Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 12
- 13. Example 5 oscillations : Doing mathematics in a physics environment is something else (At a ﬁrst year level) it is calculus... If you start with a problem "exactness" can be a handicap in some sense A small ball with a mass of 50.0 g moves on the end of a spring with spring constant 25.0 N/m. The ball is released with an amplitude of 0.300 m. A damping force F = −bv acts on egg. After it oscillates for 5.00 s, the amplitude of the motion has decreased to 0.100 m. Calculate the magnitude of the damping coefﬁcient b. Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 13
- 14. Example 5 oscillations : A small ball with a mass of 50.0 g moves on the end of a spring with spring constant 25.0 N/m. The ball is released with an amplitude of 0.300 m. A damping force F = −bv acts on egg. After it oscillates for 5.00 s, the amplitude of the motion has decreased to 0.100 m. Calculate the magnitude of the damping coefﬁcient b. Remarks: ◮ The ambiguity: what is the meaning of the amplitude? ◮ Do we start solving the problem from Newton F = ma to the second order DE and solving this with calculus which is for the students no routine at this moment, or do we start with something like ebt cos(at + δ) ◮ Frequently it is in physics common to make reasonable estimations (sloppy) if the calculations are tedious. Mathematicians don’t like that. Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 14
- 15. A slide about vectors : Interpretations for "common" subjects in physics and mathematics are sometimes different or vague? • Position 4-vector: x = (ct, x, y, z) = (x0, x1, x2, x3). • Velocity 4-vector: v = γ(v) dx/dt = γ(v) (c, v1, v2, v3). • Momentum 4-vector: p = mv = m γ(v) dx/dt = (p0, p1, p2, p3). • Expand: γ(v) = 1/(1 − v2 / c2)½ = 1 + ½(v/c)2 + ⅜(v/c)4 + … • p1 = mv1, p2 = mv2, p3 = mv3 + correction terms. • cp0 = mc γ(v) dx0/dt = mc2 γ(v) = mc2 + ½mv2 + … MOMENTUM kinetic energy rest energy = Etot owever, the combination (∆s)2 = (c∆t)2 − (∆x)2 − (∆y)2 − (∆z)2 invariant. After showing this slide, what do students think that a vector is? What is the connection to L.A.? Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 15
- 16. Something about The Axioms of Newton : If you start with a problem "exactness" can be a handicap in some sense? The ﬁrst and the second Law of Newton: ◮ In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force. ◮ In an inertial reference frame, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F = ma. (It is assumed here that the mass m is constant.) Remark: One of the questions of the exam was: What are the 3 laws of Newton? In the view of a Mathematician there are lot of things to do: What is a inertial frame? and is the ﬁrst one not a consequence of the second? Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 16
- 17. Save me: 1. It takes (a lot) of time for ﬁnding a right attitude for understanding the concepts of physics? The time slot of one quarter of a year is too short, a half year with a midterm exam will be better. 2. Doing mathematics in a physics environment is something else (it is calculus and the basics of vectors)? More experience with calculus and its interpretations is needed. 3. 14 students had passed the exam. Within Diam we also have a Double Bachelor program (a mathematics - physics combination), are they in the nearby future the only ones who can deal with physics (and modelling?). 4. Give me a helping hand... Why is Physics difﬁcult for our math students? Emiel van Elderen TUDelft June 22, 2017 17