Longer math and physics problems

[Fascheue]

When solving a math or physics problem, the process usually starts with recognizing what type of problem it is and which equations need to be used.

This is simple for more straight forward problems. If they give you mass and acceleration and ask you to find the force, you have all parts of the equation f = ma except for one.

Sometimes I find it confusing though when this is not the case, for example I had to solve a problem something like this:

There are two boats. One boat is 20 units west of the other. That boat starts moving west with a velocity of 19 units/hr. The other boat starts moving north at a velocity of 25 units/hr. What is the rate of change of the distance between the boats in 7 hours.

I eventually solved the problem, but I did so just by trying a ton of different things, many of which did not work. Eventually I set up a right triangle, with one side 20 + 19t, another side 25t, and an unknown hypotenuse. I plugged in 7 for t to get c. Then I implicitly differentiated A^2+B^2=C^2 and plugged in for all value except for the unknown dc/dt to find the answer.

What should be the thought process when solving this problem? I know how to solve it, but how do you know how to solve it? You can’t just find an equation where you have all of the known values except for 1.

 

[symbolipoint]

..., in other words, you ARE LEARNING. You improve through study and practice. You are doing this now.

What did you do to solve the example problem? Made a drawing or diagram, labeled some parts with values, expressions, numbers, picked the mathematical rules you may need to follow, wrote equations for everything you could, decided which equations you needed, and solved for unknown variables.

 

[PeroK]

When solving a math or physics problem, the process usually starts with recognizing what type of problem it is and which equations need to be used.

This is simple for more straight forward problems. If they give you mass and acceleration and ask you to find the force, you have all parts of the equation f = ma except for one.

Sometimes I find it confusing though when this is not the case, for example I had to solve a problem something like this:

There are two boats. One boat is 20 units west of the other. That boat starts moving west with a velocity of 19 units/hr. The other boat starts moving north at a velocity of 25 units/hr. What is the rate of change of the distance between the boats in 7 hours.

I eventually solved the problem, but I did so just by trying a ton of different things, many of which did not work. Eventually I set up a right triangle, with one side 20 + 19t, another side 25t, and an unknown hypotenuse. I plugged in 7 for t to get c. Then I implicitly differentiated A^2+B^2=C^2 and plugged in for all value except for the unknown dc/dt to find the answer.

What should be the thought process when solving this problem? I know how to solve it, but how do you know how to solve it? You can’t just find an equation where you have all of the known values except for 1.

Perhaps you are relying too much on the numbers. You were asked to find the rate of change of the distance between the boats at time 7. There's nothing special about 7, so why not try to find the rate of change of distance at time $t$? How do you do that?

1) You find the distance between the objects at time $t$.
a) Find the position of the first object at time $t$
b) Find the position of the second object at time $t$
c) Calculate the distance between these points.

2) You differentiate that (wrt $t$).

3) You plug in $t = 7$.

That would be my thought processes on this question - or any question that asks the rate of change of distance between two moving objects.

 

[Mark44]

Edited

0) Draw a diagram that represents the situation.
1) You find the distance between the objects at time t.
a) Find the position of the first object at time t
b) Find the position of the second object at time t
c) Calculate the distance between these points.

2) You differentiate that (wrt t).

3) You plug in t=7.

 

[PeroK]

Edited

I did do a diagram, but somehow that slipped my memory when I wrote down the steps!

 

[Mark44]

I did do a diagram, but somehow that slipped my memory when I wrote down the steps!

I trust that you did, PeroK, but this is something that beginning students are often reluctant to do, for one reason or another.

 

[FactChecker]

The fact that you presented this example as a word problem gives a clue about the possible trouble. It is too hard to formulate a solution approach while looking at the word problem. The first step is to translate the words into equations, using place-holder variables where needed. Once you see the structure of the equations and diagrams, there may be simplification steps and a general approach that will be apparent. Baby steps and practice will get it done.

 

[WWGD]

I don't think you can find general methods to approach all problems without running into Godel-type results; infinitely-many possible problems, finitely-many methods to consider.