Problems vs exercises
A problem exists when you understand what would constitute a satisfactory solution but you are unable to see how the knowledge and information you have can be used to obtain a solution.
An exercise exists, by contrast, when you know how to obtain a satisfactory solution as soon as you comprehend the given information.
Here are examples of an exercise and a problem (by the standards of the typical student of this text).
Consider this question:
An ice skater of mass m=55kg experiences a force of 20N when she is at rest due to the wind. Find her acceleration neglecting friction with the ice.
For most of you, this question is (or soon will be) an exercise. An exercise is defined as "Once you understand the question, the plan of solution is obvious." Once you understand that this question involves mass, force, and acceleration you will think of F=ma. The plan of attack is then obvious - rearrange this equation to give a=F/m and substitute the given quantities to obtain a = 0.36 m/s.
The following question should be a problem for you (if it's not, you may not get much from reading this WIKItextBook):
An ice skater of mass m=55kg experiences a force of 30N when she is at rest due to the 10 m/s wind. Find her position vs. time neglecting friction with the ice and assuming that the drag force of the wind is proportional to the square of its relative velocity.
The plan for solving this question involves
- Find the acceleration of the skater - unfortunately it involves her velocity which is unknown and is hence a differential equation.
- This is the only kind of differential equation you will need to be able to solve in this WIKItextBook - it can be solved by separation of variables
- The resulting v(t) can be integrated to find x(t)