S0.5 Problem solving
In this course we will invite you to solve many physics problems. We do so for two reasons. First, the experience will provide you with an opportunity to apply the principles of physics, and so to understand more fully what they mean. Second, the experience will also help you learn more about the art of problem solving itself –the stuff of active science.
Practising the art is the best way of learning it: but a frequent plea when confronted with a problem is – we have heard it many times – ’I don’t know where to start’. This is particularly relevant when the problems become more complex than those you might have encountered thus far in your Physics.
It’s a little like learning to drive a HGV (... I am assuming that none of you can...) When first behind the wheel, it’s sort of similar to a car, but much different as well. More levers, more gears, harder to go round corners..... What will help you gain confidence (an important skill in Physics as well as HGV driving...!) is to have a strategy that will help you get moving. With practice, you’ll be enacting the elements of the strategy automatically. Unfamiliar roads will not faze you and your rig will rumble along wherever you choose to take it.
This section presents a formal problem solving strategy that will allow you approach solving a problem systematically. We’ll identify a number of general guidelines that are helpful in keeping you on the right track. They are important for all the problem-solving you will do in this course, and beyond. So they are set out fully. After we have set them out and discussed them in general terms we shall illustrate them in action in the context of a specific worked example.
[A] The strategy - F-D-P-E-E
The strategy comprises five distinct steps......
- Focus on the problem (understand the problem)
- Describe the physics (analyse the problem)
- Plan a solution (work out a strategy)
- Execute the plan
- Evaluate the result
Let’s look at each one in a bit more detail, along with some guidelines that give you a bit more detail about what to actually ’do’
[B] Focus on the problem
Guideline 0.1 - Draw a sketch
We can often appreciate more clearly what is involved in a problem by reexpressing it in pictorial form. Most problems in mechanics cry out for a picture, since they are concerned with phenomena that we can easily imagine ‘seeing’. However even in the most abstract realms, such as quantum physics, and relativity, physicists draw pictures (symbolic, schematic) to act as problem-solving props.Guideline 0.2 - Choose a sensible notation
In most problems you will find that you need to choose symbols to respresent the important physical quantities. You will need to do this even if those quantities are given specific numerical values in the question: you will see why in Guideline 0.3. Symbols need to be explicitly defined, either by indicating their role in your sketch or in words. Choosing symbols that are simple and evocative makes life easier. Thus ‘m’ and ‘M’ are wonderful as masses but would be lousy as lengths![C] Model the problem
Guideline 0.3 - Identify the principles
Most of the problems you will meet (in this course) involve the application of just one or perhaps two basic principles. Your task is to identify which. Sometimes this will be straightforward: perhaps the problem is ‘like’ one you have seen before; perhaps there are key phrases that point you in the right direction. Sometimes you will find it harder, as you come to deal with problems that are less idealised.Guideline 0.4 - Formulate the equations
While we can go a long way with words and pictures, the full power of our physical principles is only released when we express them in the language of mathematics. Solving problems usually means translating them into a mathematical form, and invoking the tools of mathematics to follow through the consequences.[D] Plan your strategy
Guideline 0.5 - Devise a strategy
Once you’ve written down the basic equations expressing what you know, you need to take time to plan how you will manipulate them to find what you actually want to know. Time spent on this kind of route-planning may spare you the pain of finding yourself on unmade roads, at dead ends or the wrong destination. Do you have all the information that you need to be able to solve the problem?Guideline 0.6 - Do not substitute numbers until you must
Knowing the units of the terms in an equation provides an important check on whether the equation can be right: you lose this check as soon as you substitute numbers. You also lose the information about special cases: see Guideline 0.10[E] Execute the strategy
Guideline 0.6 - Use quantities in consistent units
In numberical problems, make sure than any quantities you are combining have the same units. As an example, in a kinematics problem, using a speed of miles per hour to calculate the time taken for some event will yield an answer in hours! Maybe seconds would be a better choice. In some problems, there will be no numbers specified - in this case, solve the algebra, don’t ’invent’ numbers for quantities.[F] Evaluate the answer or result
Guideline 0.8 - Ask if the answer makes sense
This is perhaps the single most important rule of the lot! It means different things in different circumstances. Sometimes it means asking whether a numerical result is physically reasonable. It usually involves other more specific tests, which we will deal with separately…Guideline 0.9 - Check the units
Our second rule governing the use of units (Key Point 0.2) provides an invaluable check which you can put into practice throughout any bit of algebraic manipulation —and always at the end. Remember that a valid equation must pass this test: if it fails, there is something wrong and it is not sensible to go further until you have sorted it.Guideline 0.10 - Appeal to special cases
Quite frequently you will find that you know some other condition that your answer has to satisfy, when you think of some special case (or limit, or range) of the physical parameters… another reason for carrying through the argument in terms of the symbols, instead of the specific values they may have in the particular instance. While satisfying this special case condition doesn’t guarantee that you’re correct, it makes it that much more likely.Guideline 0.11 - Be prepared to look from a different angle.
There are some problems which are tough –perhaps even impossible– to do if you broach them in what appears to be the ‘obvious’ way; but which become quite simple when you find a different way of looking at them. The most challenging and satisfying problems often need this kind of flexibility of thought.You’ll get plenty of practice enacting this strategy in the weekly workshops. For now, here’s a worked example to get started.....

Example
Here is a worked example showing these guidelines in action; although it anticipates some of the physics we will be reviewing later in this course, most of it (the physics) should already be familar to you. The focus here is the guidelines themselves. The numbers shown in brackets at various points in the argument refer to the guidelines.
Question
A particle of mass 2 kg sits at the bottom of a smooth upwardly curving surface. A second particle, of mass 1 kg, is released from rest on an elevated part of the surface. It slides down to the bottom of the surface, where it makes a head-on elastic collision with the 2 kg mass. The vertical distance through which the 1 kg mass moves as it slides down the surface is 0.2 m. Find the velocity of the smaller mass after the collision.
[ Take g=10 ms-2.]
Solution
- Draw a sketch (Guideline 0.1) and choose a sensible notation (Guideline 0.2)
- Identify the principles (Guideline 0.3):
- energy conservation (twice over)
- momentum conservation
- Formulate the equations (Guideline 0.4):
- energy conservation in the pre-collision motion:
- energy conservation in the collision:
- linear momentum conservation in the collision process:(0.3)mu=mv+MV
More
We have introduced some more symbols: the meanings should be obvious, but we should state them (We would expect you to!). Thus u, v and V represent (respectively) the pre-collision velocity of mass m, and the post-collision velocity of m and M. They are ‘one-dimensional vectors’ and so do not need the arrows on top (see S0.4).
Plan your strategy (Guideline 0.5):
Split problem into two sub-problems
- precollision motion
- the collision
The precollision motion:
Equation 0.1 gives
Remember the rule: do not substitute numbers until you must (Guideline 0.6).
The collision:
Rearrange Equation 0.2 to give
Rearrange Equation 0.3 and square both sides to get
Combine these two equations to find
Cancelling a common factor gives
Rearrange to give
Combine with Equation 0.4
Now feed in the numbers supplied:
(0.7)v=-2/3 ms-1Now we must ask if the answer makes sense (Guideline 0.8).
…it is a reasonable size
We must check that the units are consistent (Guideline 0.9):
…units of rhs of Equation 0.6 are
(ms-2×m)1/2=ms-1matching units of lhs- We can appeal to special cases (Guideline 0.10):
.
- If M≫m:
- If M≪m:
Help?
You may not be used to making approximations like this. In the first equation you need to realise that
It also implies that m+M≃M. The result follows.Why is this result (v≃-u) ‘sensible’? Well, if the mass M is big we can think of it as a wall, from which our particle simply bounces back.
You should make sure you see why the result v≃u for M≪m is also ‘sensible’.
- If M≫m:
Learning Resources
![]() | HRW provides a series of ‘Problem solving tactics’, spread through the text. In addition to this, we’ll be providing some dissections of (real) good and bad answers in video format online. (These will be linked from the homepage when available). |
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