Solving Problems in the Physical Sciences
Many students find mathematical
problem-solving to be the most difficult aspect of a physical science
course. It is no wonder, given that
word problems of this type demand both the mathematical and the linguistic
centers of your brain to work together.
On the plus side, by developing your ability to do this type of
problem-solving you can improve your performance not just in physical science,
but also in mathematics and economics, and on the SAT.
The Method
1. Read the question
carefully and consider the description of the problem and the data given. Also check for any data tables, or
information mentioned in class as “basic knowledge,” for example: the
acceleration due to gravity is 9.8m/s2. In many problems it is useful to draw and label a diagram to help
you visual the scenario.
2. Figure out what quantity
the question is asking you to calculate.
3. Decide which concepts,
principles and equations apply to the problem.
4. Considering the data
given, the quantity requested and the applicable principles/concepts, choose
the equation(s) that can be used to solve for the desired result.
5. Manipulate the
equation(s) to arrive at an algebraic solution for the requested quantity.
6. Check to see if the data
is given in the appropriate units. If
it is not, convert to the appropriate units. (If using vectors, choose an
appropriate coordinate system and analyze the data using it.)
7. Substitute the data into
the formula and calculate the result.
8. State the answer with
correct scientific notation (if used), units and direction (if a vector). (For 11s and 12s, consider significant
figures.)
An Example
Question: A jar of volume
1.000 L is to have water of mass 1500 g poured into it. How much, if any, water will overflow the
jar?
1. The data given is: volume of jar = 1.000 L and mass
of water = 1500 g, but you will also have heard in class that the density of
water is 1000 kg/m3. (In an
exam it would be in a data table at the end of the exam.) The basic idea of the problem is that if the
volume of 1500 g of water is greater than 1.000 L it will overflow by whatever
the difference is between the two volumes (if the water volume is greater).
2. Quantity to calculate: volume of 1500 g of water
and its difference from 1.000 L if it is greater.
3. Equation(s): mass density = mass/volume
4. Choice of relevant equation: only the one in this case.
5. Algebraic solution: volume = mass/mass density
6. Units: the mass is in grams but the density
is in kg so we must change the mass to match.
1500 g [1kg/1000g] = 1.500 kg.
7. Calculation: volume = 1.500kg/1000 kg/m3
= 1.500 x 10−3 m3.
To compare this to the 1.000 L jar I must convert the 1.000 L to m3 (Again, I would have mentioned in
class that 1000 L = 1 m3 and this would be in the data table on an
exam). 1.000L [1m3/1000L] =
1.000 x 10−3 m3.
The water is indeed of greater volume and the difference in volume is 1.500
x 10−3 − 1.000 x 10−3 = 0.5 x 10−3
8. Answer: 5.00 x 10−4 m3
Rubric for Problems
A set rubric for problem-solving on assignments and exams is difficult, but the following will give you some guidelines:
Understanding concepts: If you display understanding of the concepts by, for example, choosing the appropriate equation(s), or applying a principle correctly, you will receive a 1 to 2 marks depending on the level of difficulty of this step for that particular problem.
Mathematical solution: If you need to rearrange formulas and do so correctly you will receive a mark or two depending on the level of difficulty.
Choice of quantities: Choosing the correct quantities from the question or tables of data to substitute into the formula is worth another mark. Two marks if unit conversions are involved.
The answer: Writing out the answer with correct units and direction (where appropriate), and with correct significant figures (for grades 11 and 12) will get you another mark. If you only write out the answer and show no working you may receive full credit, but if the answer is incorrect and you have shown no working you will receive NO CREDIT.
Total marks: These will range from 4 to 7 in general, but may be greater for complex problems made up of two or more calculations. In that case, each calculation would be marked using the rubric. Incorrect answers in one step will, of course, lead to the incorrect final answer but marks will only be taken off for the first mistake.