In this section we make some suggestions on how to approach this course. Individuals are all different, and not every suggestion will apply equally to everyone. But in our experience, most suggestions will apply to most people.

- Plan on spending a lot of time -- a minimum of three to five hours of study for each hour in class, not including the time you spend on labs.
- Physics texts cannot be read quickly. You may find an initial skimming of a section or chapter useful, but by and large you should read with paper and pencil in hand. Don't just read derivations, but work them through yourself, and try to understand every step. Try both to develop an intuitive, qualitative grasp of the material, and to understand how the intuitive picture is expressed and developed mathematically.
- Treat the worked examples in the text and study guide in the same way -- work them through. When you're done, try closing the book and seeing if you can work out the example without help. Also, try explaining the example to yourself or someone else -- not just the mathematical steps and equations, but why you are using those equations, and what physical principles are involved.
- Work large numbers of problems; the assigned homework problems should be regarded as a warm-up. You should do a substantial fraction of the problems in the text and study guide for each chapter. There are at least two reasons for this extraordinary demand. First, working problems helps you understand the physics more thoroughly; it is a common experience (among physicists as well as physics students!) to read through new material carefully, think you understand it, and then be stumped when you try to do a calculation. Second, there are specific techniques that are involved in solving problems. We will lay out some general guidelines below, but the more practice you get, the more you will improve.
- Work with others. It often helps to study and work problems with one or two others in the class. And by working in groups, you can often work more problems, and cover more material, than you can working alone.
- Don't get behind in your work. Physics is a cumulative subject; the material you will study later in the semester (and the year) builds on what you have learned earlier.
- GET HELP WHEN YOU NEED IT. We expect you to have a lot
of questions about this material, and to experience
a certain amount of confusion and frustration at
times. It is a necessary, if unfortunate, part of learning. One
of the most important functions of the faculty at
St. John's is to be available for individual help;
in addition. The only stupid questions are the ones
that don't get asked.
In addition, some tutoring help is available on both campuses. See your instructor or the appropriate counseling office for details.

- State the equation in words, and try to understand what the equation is saying about nature. Learning an equation is NOT the memorization of symbols, but the understanding of a physical process.
- Describe each quantity in the equation. What does the quantity describe physically? Is it a vector or a scalar? What are its units? What sign conventions apply?
- Describe each operation in the equation (e.g., differentiation, scalar or vector multiplication, summing, etc.) How would you actually perform these operations given all the quantities?
- Describe the relationships implied by the equation. How does each quantity depend on the others? Do these dependencies relate to your own experience?
- What is the origin of the equation? Is it a definition (for example, momentum as the product of mass and velocity) or a relationship between independently measurable quantities? If the latter, what experiment could check the validity? Does it represent an empirical law, or is it derived from more fundamental laws? Do you understand the derivation?
- Try to think of a simple physical system described by the equation.

- First, get an intuitive idea of what the problem involves. Describe the problem to yourself. Make a careful sketch. Ask yourself what the qualitative features of the solution are likely to be. Learning physics involves combining an intuitive understanding and a mathematical description of nature. If you jump too quickly to the mathematical description, it is easy either to head off in a wrong direction, or to get a result that you don't understand fully -- even if it's correct.
- Once you understand the problem intuitively, plan a solution. What information are you given, and what do you need to calculate? What physical laws pertain to this problem?
- Now, proceed to a mathematical description of the problem. What equations will you need? How will you use them? Try not to think of the equations as "formulas", but rather ask yourself what they tell you about nature. (See the discussion above on understanding equations.)
- Carry out your calculations algebraically at first; that is, don't put in numerical values until the end. If you substitute in numbers too soon, it is easy to lose sight of how various quantities affect each other. Suppose, for example, that you are considering a ball tossed into the air. How does the maximum height depend on the initial velocity? If you put in numbers too soon, you can easily lose sight of relations of this sort, which are often the main point of the problem!
- Finally, check your answer. Is it "reasonable," both numerically or algebraically? Or have you, for example, calculated a velocity that is faster than light, or inadvertently predicted that a ball thrown into the air will accelerate upwards? Checks of this sort can either give you confidence in your solution, or point up possible mistakes.

All of this, of course, is more easily said than done. As you begin studying physics, both your intuition and your mathematical skills are relatively undeveloped; it can be hard to know what to trust, or where to begin. Nor can the above suggestions be followed mechanically, or by rote. Try things. You may think you know the right equation, but aren't sure. Try it. You may get the "right" answer; so far, so good. But don't stop there! Go back and understand the equation, where it comes from, how it's derived, what it says about nature, why it applies to this problem. And get help if you need it. Then you'll have learned some physics.

As an example of this problem-solving strategy, consider the following problem, which is typical of the ones we will be doing early in the semester.

**Problem:** A ball is thrown vertically upward with an initial
velocity of 5 m/s.

- What is its velocity after 0.5 seconds?
- What is its velocity after 1.0 seconds?
- How long does it take to reach its maximum height?

**Step 1:** Examine the qualitative features of the motion.
The only force acting on the ball is the force
of gravity, which near the surface of the Earth produces
a constant acceleration. That force will act
to slow the ball, bring it to a stop at some
maximum height, and then speed it downwards; the
speed will continue to increase until the
ball hits the ground. (This sort of analysis
is fairly simple here, but it will become more sophisticated as we
encounter more and more challenging problems.)

**Step 2:** We know the initial velocity (call it *v*_{0}) and the
constant acceleration of gravity *g*. Since the
ball can move both up and down, we need to choose
a sign convention; arbitrarily we will choose
the positive direction to be up (any choice is
OK, as long as you're consistent). We don't know how long
the ball will take to reach the top of its
arc, or what the velocity will be at any
instant; so we must look for a mathematical description that
relates *v*_{0} and *g*, which we know, to the
(unknown) velocity at any instant.

**Step 3:** An appropriate equation would seem to be

*v* = *v*_{0} - *gt*.

Notice that at *t* = 0, *v* is positive, so our sign convention
is obeyed. Since the force of gravity is downward,
we have "-*g*" for downward acceleration.
(How would we write this equation if we
had chosen down to be positive?) At this point,
you should ask yourself if you really understand
this equation. Where does it come from? How is
it derived? Does its form depend on the fact
that the acceleration is constant? How would the
equation look if the maximum height were near the
Moon's orbit, where the acceleration of gravity
is much smaller?

Note also (step 5) that the equation agrees with our
qualitative analysis of the problem. As the time
increases from zero, the velocity, which is
initially *v*_{0}, decreases to zero and then
increases in a negative sense (i.e., is directed
downwards).

Note also that the equation is seriously misleading in one way. As time gets larger and larger, the equation predicts an increasing downward velocity. But this prediction is wrong! Eventually the ball will hit the ground, and either come to rest or bounce up again. Even in a simple problem, apparently, we can come to grief if we use the mathematics blindly.

**Step 4:** Now let's get some numbers out. For part a,

*v* = 5.0 m/s - 9.8 m/s^{2} (0.5 s) = 0.1 m/s (+ hence up).

The ball is apparently close to the top of its arc here,
since the velocity is small compared to *v*_{0}, but
still positive. Note too that the units are
correct. Get used to the habit of checking
units; as problems become more complicated,
analysis of units can be a helpful check. Part b, of
course, is very similar:

*v* = 5.0 m/s - 9.8 m/s^{2} (1 s) = -4.8 m/s (- hence down).

Here we must be careful. Do we know the ball hasn't hit the ground yet? How? What would the situation be at 2 seconds? 60 seconds?

Now consider part c: We know the velocity must be zero at the instant that the ball reaches its maximum height; hence

0 = *v*_{0} - *gt*;

we can solve this equation for its only unknown, the time of maximum height, to obtain

*t* = *v*_{0}/*g* = 5.0 m/s / 9.8 m/s^{2} = 0.51 s.

This result is certainly consistent with part a. Does it tell us any more about part b? Again, note that the units work out.

**Step 5:** In this instance, we have done our checking as we
went along. We checked units, and also looked
for internal consistency. If for example we had found
*t* to be 0.45 s in part c, we would have the
ball still going up after it had reached its
maximum height. What other checks of this sort occur to you?