11+3 tips to solve physics problems
Practically all the exercise books I have (publishers send me some from time to time) are all the same: only the set of exercises changes. The same exercise, in fact, can be dressed in a different way and, therefore, they are not so different either. I must confess that I do not like any of them.
Rather then teaching how to solve a problem, the chapters of these books usually open with a reference to the fundamental formulas concerning the subject dealt with, and then propose a series of exercises, of increasing complexity, followed by their solution, which are almost always presented in a very clean, plain, natural and obvious form. The correct mathematical expressions are identified without even discussing them, and, after a bit of algebraic manipulation, the solution is served.
Reading such books, solving physics problems seems a very simple job, indeed. Nobody suspects that the solution of a physics exercise requires much more time than reading its solution. The result is that the student feels frustrated and depressed.
Indeed, the way to the solution is almost always not straightforward; first of all we have to find the main road, because, often, several of them open to our eyes. Then, we need to know when to turn back, if necessary, if we realise we have taken the wrong road. We need to take a break from time to time to see if we are going in the right direction and if we have not made any mistakes.
In my view, a good exercise book should not just illustrate the solution; it should instead suggest strategies, show how and where you can go wrong; highlight the points where you need to pay attention, etc. It should be, in short, more the story of an adventurous journey, including episodes where the protagonist falls in a quagmire and has to get out of it, rather than the placid chronicle of a walk in the park.
In this (long) post I try to outline some elements I usually repeat when I solve exercises for my students.
Phase 1: reading the exercise
The text of an exercise is always written very carefully. This is, in a way, also a limitation of the exercises, because it denies the student to make his own hypothesis about whether or not to make certain choices, but I don’t want to add too much entropy here, so let’s analyse the typical case.
Every single word in the text has a meaning in the economy of the exercise. If, however, one starts immediately to try to outline one’s solution strategy, one risks losing sight of the overall picture.
Rule no. 1: Read the text of the exercise as if you were reading a news article in a newspaper. Ignore the details. Do not dwell on the details. Get a mental picture of what is described in very general terms.
In this way you will have a fairly precise idea of the situation described in the exercise.
Rule no. 2: Read the text a second time, slowly. This time dwell on every single expression in the text and try to understand if there is any relevant information (almost always so).
For example, let us consider the following exercise (taken from J. Walker’s book and appropriately modified by me):
A train is travelling up an inclined slope of 3.73° at a speed of 11.7 km/h, when the last car detaches and starts to proceed by inertia without friction. [questions follow]
The relevant and implicit information are: a) that the protagonist of the problem is not the train, but the last car; b) that the latter can be assimilated to a point-like particle, because no details on its shape are given; c) that initially it moves at the same velocity as the train to which it is attached; d) that, at a certain point, the constraint that keeps it hooked to the train is missing and, therefore, it freely falls (but let us not forget that its initial speed is upwards). The text explicitly mentions the absence of dissipative forces.
Phase 2: making a sketch
Rule no. 3: while reading the text the second time, make an outline of the systems involved in the problem and assign a symbol to each piece of information, giving it the value provided by the problem. Do this at the top of the page and do not write anything else in this space.
In the example, we have to draw an inclined plane on which we will draw, in a stylised way, the car (not the train, which it is not interesting, because it will continue to move in a uniform rectilinear motion, so its state does not change).
Sub-rule 3A: When drawing an inclined plane (generally when you have to draw something in the shape of a right triangle), always draw it so that the two sides of the triangle are very different from each other, unless differently specified.
This helps to identify angles. On average, students tend to draw an inclined plane as follows
In this way they are always in doubt about which of the two possible angles highlighted in the drawing is equal to θ. Drawing the triangle like this, instead,
you can immediately see which angle corresponds to which other one.
Sub-rule no. 3B: if the exercise requires to identify a reference system, choose one in which one of the axes is parallel to the initial or final speed of the system.
This simplifies the equations of motion and, in the most fortunate cases, up to two of them can be eliminated.
Sub-Rule no. 3C: when writing the values of data provided by the problem, immediately transform them into SI units. It may not help, but it doesn’t hurt and can save you from gross errors due to distraction or haste.
In the example, we would write v = 11.7 km/h = 3.25 m/s. If our calculator is set to use angles in radians, we will also immediately transform this quantity: 3.73°=0.0651 rad. If one of the data provided by the problem is a diameter, calculate and write down the corresponding radius immediately.
Phase 3: identifying the strategy
Here comes the most difficult part: identifying the solution strategy. You can consider the following tips.
Rule no. 4: identify the topic, among those possible, by trying to identify the chapters of the book where similar situations are discussed.
In the case under consideration, one can think that the relevant issues are those related to Newton’s Laws, because we are talking about the motion of something subject to the action of one or more forces (gravity in the example).
Rule no. 5: Write down the most important and general physical laws on the subject that come to mind. Highlight those in which the physical quantities present in the text, or emerging from the diagram you have drawn, appear.
In this case we would write a=F/m, identifying in F the vector sum of the gravitational force and the normal force exerted by the plane. This second force must come to our mind because, if not present, the body would follow a trajectory defined by the direction of gravity only. This is an important relation because it contains the expression of the force we have drawn in the scheme. In this case the speed does not appear, but it is obtained knowing that the motion will be accelerated with constant acceleration, therefore v=v(0)+at.
The problem continues with two questions:
a) After how long does the car temporarily stop?
b) How far does the car travel before stopping temporarily?
The first question suggests that the car must stop at a certain moment. Therefore, its speed must become zero. Having written that v=v(0)+at, we understand that we have to take advantage of this relation by imposing that v=0. It will be enough to correctly identify the expression of a from the relation we have written, that is a=F/m, F being the component acting in the direction of the plane and that, looking at the drawing, is mg·sin(θ). The mass thus disappears from the equations, which indicates that we are on the right track because we do not know it. The result is an equation of the type v(0)+g·sin(θ)t = 0, that can easily be solved to obtain t.
Phase n. 4: computing the final result
Rule no. 6: Compute the numbers only at the end, when only quantities whose value is known appear on the right of the = sign.
Do not forget to do a dimensional check before replacing the numbers. If you have arrived at the solution using only the symbols, this can be done, otherwise it is impossible.
Rule no. 7: Before using the numbers to find the solution, check the dimensions. If they are not correct, you have definitely done something wrong.
If, as in the example, the dimensions are correct, we can move on to the numerical solution. If you have followed the Sub-rule 3C, you will not need to indicate the units of each data: they will necessarily be consistent and expressed in the same system of units. Therefore, the result will also be expressed in the units used in the SI.
Rule no. 8: Replace the symbols with numbers, without the need to indicate the units (if you have followed Sub-rule 3C). Indicate the units of the result according to its physical quantity, in the same system of units.
In this case the time will be expressed in s.
Rule no. 9: Approximate the result, if necessary, using a number of digits consistent with the number with which the data is provided in the problem.
Speeds, for example, are given with three digits, like angles: the result is therefore written using three significant digits.
The distance travelled by the car can be calculated by recalling the expression of accelerated motion. In fact, in this case, the car moves with some initial speed, subject to constant acceleration directed in the opposite direction.
Rule no. 10: When the problem describes an initial and a final state, it should immediately occur to you to exploit the principle of energy conservation.
For the first question, this is not possible, because the problem explicitly mentioned a time. In this case the energy does not help. For the second question, however, we know that initially the car moves at speed v and is at a certain altitude. In the final state, instead, it is at a higher altitude, but its speed is zero. So we can write that the initial energy is entirely kinetic, assigning to its gravitational potential energy the null value: E=½mv². In the final state the energy is all potential and E=mgh. Having the energies to be equal, we will have that h=v²/2g. Clearly, h is the difference in altitude and therefore is equal to the distance travelled along the inclined plane (the hypotenuse of the triangle) times the sine of the angle. Thus we will find the answer to the question.
By exploiting energy conservation, only a first degree equation must always be solved. On the contrary, if the law of motion is used, a second-degree equation must be solved (at least), which can also be relatively complicated and can, therefore, easily lead to mistakes.
So, if you can clearly identify the initial and final state, you should not even think to try a different solution. This is true in any situation and in every physics topic.
Rule no. 11: Always evaluate whether the numerical result obtained is reasonable. If it is not, look carefully for possible mistakes.
In our case the time needed for the car to stop before sliding down from the inclined plane is a little more than 5 seconds. If we had obtained 5 ms or 510 s we would have had to worry, because too little or too long a time is in contrast to our experience. If the data of the problem appears reasonable, so should the solutions.