LINEAR MOTION

I will use x for distance, v for velocity, a for acceleration, t for time. This is fairly standard except that people use all kinds of things for distance: x, y, d, s, h and r are the most common. When people say linear motion, they usually mean not only motion in a straight line, but also the special case of velocity vs time being a straight line graph. This is the sense used here.

Using the definitions of velocity and acceleration, your physics text and your instructor will derive the set of linear motion equations: x = v_o t + (1/2)at² , etc. At first, maybe you barely know the meaning of acceleration and are therefore in a fairly confused state. However, the thing to do is work a reasonable number of problems using these equations and you will be OK.

Important point:  The equations apply only to cases of constant acceleration.
If the graph of v vs t is not a straight line, you may get into trouble.

To solve these problems, first think about what is happening. Sometimes you can science it out just by common sense. Predict what will happen before calculating anything. Pay attention to the units. Are they compatible? (Mi/hr might have to be changed to ft/s, for example.)

Finally, write down the value of each quantity given including those stated indirectly (for example if the car comes to a stop, they don't need to state that the final velocity is zero). Pick an equation that has in it your known quantities and the thing you want to find. Do the algebra and solve.

There is a Christmas card that goes like this:
ABCDEFGHIJK_MNOPQRSTUVWXYZ (translation: NOEL).

What does that NOEL have to do with anything? Well, when you write the linear motion equations for the case of x_o = 0 and t_o = 0, each equation has one of these quantities missing: x, v_o , v, a, t. Thus

1. x = (1/2)(v_o + v)t------- no a

2. x = v_o t + (1/2)at² -------no v

3. x = vt - (1/2)at² ---------no v_o

4. v = v_o + at ---------------no x

5. v² = v_o² + 2ax -----------no t

(Authors of texts frequently leave out #3 because it is used less often.)

What if initial x and t are not zero? (You can define them to be zero if the problem concerns one body, but maybe not for a 2 body problem.) Then replace x with 
(x-x_o ) and t with (t-t_o ) in the above equations.

Do enough problems to cause you to remember the equations. Do not sit and memorize them, but know them.

You need to nurture your equations. Work with them to show that you care. Love them, but don't coddle them; be firm. Make them do chores to develop their capabilities. If you do not handle them right, they will probably rebel at the worst time, like during a test.

DERIVING: I mentioned that the five equations are derived from the definitions of v and a. I lied. There is one more thing you need: when numbers in a list increase (or decrease) uniformly (like 5, 8, 11, 14, or 2.3, 2.4, 2.5) the average can be found by adding the first and last and dividing the result by 2. This is true also for something changing continuously, like velocity. Thus the average v is (v_o + v)/2. You need this to derive equation 1.
Equation 2 and 3 are found by writing equation 1, then replacing v with v_o + at to get equation 2, and replacing v_o with v-at to get equation 3.
Equation 4 is simply a rearrangement of the definition of acceleration, a = D v/t.
The easiest way to derive equation 5 is to multiply the average v by the change of v. Put this on the left: (1/2)(v+v_o )(v-v_o ). But the average v is x/t and the change of v is at, so their product is ax (the t cancels out). Put ax on the right. Then with a little algebra you will get equation 5.

GRAPHING: The most useful graph is v vs t. The slope is the acceleration, and the "area" under the line is the distance. I call it "area" with quotes because it has units of velocity multiplied by time, which is distance. All physics texts discuss these ideas. Be sure to become familiar with slopes and areas. Play with x vs t and a vs t graphs also. The v vs t graph is a straight line (if not, divide it up into a number of straight lines-- lots of them if it is a curve). x vs t will be a parabola if there is constant
non-zero acceleration.

When you get stuck on a problem, sketch a graph how you think v vs t should be. Write down what you know and what you want to find. Be on the lookout for things which are not stated explicitly. For example if the problem is about an object which was dropped, this implies v_o = 0.

Go back using your browser's back button. Do not click on back or you will be cursed with nonlinearity.

My main physics pages:

mechanics
fluids, heat, electricity and magnetism
vibrations and waves
quantum

Comments, questions: fredrick.gram@tri-c.edu