# An Algebraic Approach (AP Level)     [Chapter 2 Objectives]

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In many of the situations that we will study in mechanics, the acceleration of the object of interest will be at least approximately constant. This turns out to be fortunate, because constant acceleration allows you to analyze the motion using only simple algebra.

## How to Solve Kinematics Problems

I'd like to say that solving numerical kinematics problems is highly creative problem solving at its best - but that would be a lie. Although beginning physicists often find kinematics problems quite challenging at first, once you get the idea, they are generally pretty routine.

In order to get into the rhythm of solving kinematics problems as quickly and easily as possible, use the following steps to guide your problem-solving efforts. This method is illustrated in several examples on the following pages.

Remember that there will be many situations in physics in which your solution to a problem will be graded - not just the answer. In fact, in many cases, the solution will be weighted much more heavily than the correct answer. Therefore, it is important to develop a professional problem-solving style. Here's a good way:

1. Understand the Problem:
1. Read the problem carefully. What do you need to find? You would be amazed at how many people panic and try to start solving a problem without really knowing what they are supposed to do!
2. Make a sketch that shows the problem situation. In particular, most kinematics problems deal with an object moving from some initial situation to some other ending situation - a car starts from rest and travels down a road, or a ball is dropped and falls toward the Earth, for example. Be sure that your diagram represents both the starting and ending situations. Don't spend time making your sketch artistic - concentrate on the physics.
3. In your sketch, indicate what you know about both the starting and ending situations. Indicate the direction of each kinematics quantity - displacement, velocity, and acceleration.
2. Translate the Problem into "Physics Language": Make a list of each kinematics quantity in the problem. Identify each one with the proper kinematics variable. Don't forget units!
3. Find a Kinematics Relationship that Will Solve the Problem: Do a "mental scan" of your list of kinematics equations to find the equation that contains the kinematics quantities in your problem. Write this equation down. Of course, you want the equation that will solve the problem as efficiently as possible, but, as you will see from the examples, there is more than one way to solve most kinematics problems.
4. Solve: Solve the mathematical equation for the kinematics quantity you need to know.
5. Substitute Known Values: Substitute the known values of the kinematics quantities into your equation. Treat the units as you would any other algebraic quantity. Don't copy down every digit on your calculator display! Round your answer to the same number of significant digits as the least precise value in the problem. Don't forget units!
1. Note: Professional physicists and engineers would solve, algebraically, for the quantity they want to know first, and then substitute the known values, as shown in steps 4 and 5. If you would be more comfortable substituting the known values first, and then solving an equation with just one variable in it, that will be fine. You can use whatever method you are comfortable with.
6. Report Your Answer: Don't expect someone to find the answer to the problem in your solution and interpret it correctly. Write a sentence that gives the complete answer to the problem. Be sure to mention direction, if appropriate, and never forget proper units.
7. Check: Take a minute to be sure that you have actually answered the original question, and that your answer is reasonable. If the problem has multiple parts, be sure that your answer to this part is consistent with the rest of your solution.

## Translating Verbal Kinematics Problems Into "Physics"

In order to apply the power of mathematics to kinematics problems, you need to be able to translate a verbal description of a kinematics problem into the symbolic language of mathematics.

A kinematics problem generally consists of three situations. In the original (or starting) situation, some object is located at position "xo" and has velocity "vo" at the instant that the clock reads "to". Then, the object has a constant acceleration "a" for some time " " while it moves a distance " ". In the final situation, the object is located at position "x" and has velocity "v" at the instant that the clock reads "t". Given values for some of these quantities, your job is to find values for some of the others.

The table below shows the variables that we will use to represent each kinematics quantity.

Customary Symbols for Kinematics Quantities

Kinematics Quantity

Mathematical Symbol

Position:

final position - where the object is located at the instant the clock reads "t"
x
original (or starting) position - where the object was at the instant the clock read "to"
xo
displacement (signed, net) distance the object moved Velocity:

final velocity - the object's speedometer reading at the instant the clock reads "t"
v
original (or starting) velocity - the object's speedometer reading at the instant the clock reads "to"
vo
change in velocity average velocity Acceleration:

(constant) acceleration
a

Time:

t

original (or starting) clock reading

to

time interval (change in time) ## Kinematics Equations:

In the table below, you will find the 7 equations that you need to know to solve kinematics problems. Don't panic! The first 3 equations are definitions that you have already seen! Notice that the equations in the table have been divided into two groups. The "green group" are mostly kinematics definitions. Using these equations often leads to multistep solutions of kinematics problems - but each step is pretty simple. The "blue group" are more sophisticated equations that can often solve kinematics problems in one step, but the algebra involved in using them is often slightly more complicated. The following pages will show you where these equations come from, and show how the equations can be used to solve kinematics problems.

The Equations You Need to Know to Solve Kinematics Problems

(Memorize These Equations)

Kinematics Equation

Comment The definition of displacement in algebraic symbols- Note that " " always means "change in", and anything always means "final value - original value". Therefore, defines "change in velocity" and defines "change in time". This is the definition of average velocity in mathematical symbols. (A bar over a quantity denotes "average".) Velocity is the rate position changes. Average velocity is displacement divided by time.

This equation is often seen in the form , which says, essentially, "distance equals average velocity times time" This is the definition of acceleration in mathematical symbols. Actually, we said that average acceleration is change in velocity divided by time, but since acceleration is constant, average acceleration and instantaneous acceleration are the same. Here are some examples of its use in solving kinematics problems. This equation says that if the acceleration is constant, the average velocity in any time interval is simply the average of the original and final velocities. It is very simple to use and handy in simplifying many calculations. Its derivation requires calculus, though. This equation is the good old "distance equals average velocity times time" you learned back in 5th grade - dressed up a bit! It can be very useful, particularly in conjunction with the previous equation. This equation is very closely related to , and often either one can be used to solve a problem. Click here to see how this equation is used, and where it comes from. This equation is useful in many situations. Here are some examples, and here is its derivation. In many kinematics situations, you know speedometer readings, acceleration, and distance, but you don't know the time interval involved. This equation comes to the rescue in this situation. Here are some examples.     [Chapter 2 Objectives]

BHS -> Staff -> Mr. Stanbrough -> Physics -> Mechanics -> Kinematics -> this page

last update November 10, 2007 by JL Stanbrough