**Equations**

Students generally have one of three attitudes toward equations:

- sheer hatred (enough said),
- cold pragmatism (plug in numbers and get an answer), and
- warm fondness.

Try adopting the last attitude. Many students do not realize that equations are merely a way to contain useful information in a short form. They are sentences in the concise language of mathematics. You should not have to memorize most equations in the text, because by the time you learn each chapter, the equations should feel natural to you. They should feel like natural relationships among familiar quantities.

For example, consider one of the first equations you ever encountered, distance equals rate times time, that is

Δ

*x*=*v*Δ*t*(1)It makes sense that, in a given time, we can go twice as far if we go twice as fast. Thus ∆

*x*is proportional to*v*. On the other hand, for a given speed, we can go twice as far if we travel twice as long a time. Thus ∆*x*is proportional to ∆*t*. We would never be tempted to write*v* = Δ*x*Δ*t*

or

Δ*t* = *v*Δ*x*

because these equations give relationships among the quantities that we know to be wrong. Note also that the units work out correctly only in equation (1).

Another example is the second law of motion, which we will encounter in Section 3.B. If an object has a single force on it, then its acceleration is proportional to the magnitude of the force and inversely proportional to its mass. Instead of words, we simply write

*a* = *F*/*m* (2)

Now let’s think about the equation. What would we do if we forgot it? If we stop to think about it, we could figure it out. First, we know that force, mass, and acceleration are connected somehow. If we have two objects of the same mass, and we apply three times as much force to the second object as to the first, then we have a picture like that in Figure 1-1. The greater force causes the greater acceleration, so we can guess that they are proportional. We write

*a* ~ *F*

Figure 1-1

If we apply the same force to two objects of different masses, then we expect the smaller object to accelerate more (Figure 1-2). Thus we can guess that the acceleration is

*inversely*proportional to the mass, so we write*a* ~ 1/*m*

Figure 1-2

Combining these two proportions we get

*a*=

*F*/

*m*

as in equation (2).

When you take the MCAT, you really should have the equation

*F*=

*ma*in your head, but if you train yourself to think this way, it will be easier to keep the formulas in your head. This will make it possible to recover the formula if you forget it. And you will understand physics better. Most importantly, you will be better able to apply the concept behind the equation.

Some equations are a little more complicated. An example is Newton’s law of gravity, which gives the force of gravity between two objects:

(3)

where

Well, start with the idea that objects with more mass have a greater force of gravity between them, so write

*G*is a constant,*m*_{1}and*m*_{2}are masses of objects, and*r*is the distance between them. How would you ever remember this equation?Well, start with the idea that objects with more mass have a greater force of gravity between them, so write

Also, if objects are far apart, the force of gravity between them is less, so write

There is a constant, so write

The only part that needs to be memorized is the “square” in the denominator, so that we have

That’s why we call gravity an inverse-square force.

The MCAT will not ask you to substitute into an equation like equation (3), but it may ask a question like, “What happens to the gravitational force between two objects if the distance between the objects is increased by a factor of four?”

We can tell from equation (3) that an increase in distance results in a decrease in force, because

If this last point seems opaque to you, try some numbers on a more familiar equation, such as that for the area of a circle

The MCAT will not ask you to substitute into an equation like equation (3), but it may ask a question like, “What happens to the gravitational force between two objects if the distance between the objects is increased by a factor of four?”

We can tell from equation (3) that an increase in distance results in a decrease in force, because

*r*is in the denominator. Because the*r*is squared, a factor of 4 in*r*will result in a factor of 4^{2}= 16 in F_{grav}. The answer is that the gravitational force decreases by a factor of 16.If this last point seems opaque to you, try some numbers on a more familiar equation, such as that for the area of a circle

*A* = *π**r*^{2} (4)

What happens to the area when the radius increases by a factor of 3? (Answer: It increases by a factor of 9.) Try it with

Another equation is that for the surface area of a sphere

*r*_{1}= 4 m and*r*_{2}= 12 m, or with some other numbers.Another equation is that for the surface area of a sphere

*A* = 4*π**r*^{2} (5)

What happens to the surface area of a sphere when the radius increases by a factor of 3? (Answer: It increases by a factor of 9. Surprised? What about the factor of 4? Try it with

Another example concerns the volume of a sphere

*r*_{1}= 4 m and*r*_{2}= 12 m.) The surface area of a sphere is an equation that you just have to memorize. It is difficult to get an intuitive grasp why the 4π should be there. On the other hand, the*r*^{2}is natural in this equation. Why? (Think about units.)Another example concerns the volume of a sphere

(6)

What happens to the volume when the radius is doubled?

In this chapter we discussed the importance of units in solving problems. If a problem involves only simple proportionalities and there are no unitless proportionality constants, then we can obtain a quick solution simply by keeping track of units.

The example in the text demonstrates all the techniques involved.

We also looked at equations as the language of physics. If you read equations as sentences containing information for you to understand, then the equations will seem less foreign than if you look at them as abstract collections of symbols. Each time you encounter a boxed equation in this book, you should spend some time thinking about what the equation means.

The example in the text demonstrates all the techniques involved.

We also looked at equations as the language of physics. If you read equations as sentences containing information for you to understand, then the equations will seem less foreign than if you look at them as abstract collections of symbols. Each time you encounter a boxed equation in this book, you should spend some time thinking about what the equation means.