Kinematics: Motion in One Dimension

Using the Mathematical Model in Problem Solving

Now you are familiar with the terminology associated with our mathematical models, let’s look at some examples of how to apply that model. You need to remember the “rules” associated with it, so you know when to use it in calculations and when to use a different model. You should also be familiar with how algebra works and be able to manipulate this model to find any of the variables associated with it. In science, these are called derived quantities.

Key Concept Recall Mathematical Models

Average Speed
v equals d over delta t

Average Velocity
v equals delta x over delta t

Position of Object (Constant Motion)
x equals v times t plus x nought

Note: the second and third equations are exactly the same, just rearranged.

Let’s look at some examples.

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Click here for some techniques for problem solving.
It is a good idea to establish some good problem solving habits now. These first few problems may not need this long “process,” but it will come in handy later when the problems get more difficult. Start out with listing all the given information, listing the variable and the value of the quantities. Then, list the variable that is unknown (what you are trying to find). Draw a diagram to help visualize the problem. Also, indicate the equation (mathematical model) that will help you solve the problem. The equation will have the known variables and the unknown variable(s), but nothing more. Sometimes you will have to do an intermediary calculation in order to use a particular equation. Keep this in mind as you list the equations you will use. Lastly, you can plug the values into the equation and solve for the unknown value. Use the following structure on your paper:

Given:
List given variables with values		 Diagram:
Draw a diagram that illustrations the problem (if necessary)
Find:
List variable(s) to find		 Equations:
List equations to use
Solution:
Work the problem here

Example 1: Average Speed

What must be your car’s average speed in order to travel 235 km in 3.25 h?

Given:
d = 235 km
t = 3.25 h
Find:
speed, v		 Equations:
v equals d over delta t
Solution:
v equals d over delta t equals two hundred thirty five kilometers over three point two five hours equals seventy-two point three kilometers per hour.

The car’s average speed is 72.3 km/h. Since there is not a reference to unit within the question, we will leave the answer as km/h.

Example 2: Average Velocity

A rolling ball moves from x1 = 3.4 cm to x2 = -4.2 cm during the time from t1 = 3.0 s to t2 = 6.1 s. What is its average velocity?

Given:
x1 = 3.4 cm
x2 = -4.2 cm
t1 = 3.0 s
t2 = 6.1 s
Find:
average velocity, v		 Equations:
v equals delta x over delta t
Equations:
v equals delta x over delta t equals x two minus x one all over t two minus t one equals negative 4.2 minus 3.4 all over 6.1 – 3.0 equals negative 7.6 over 3.1 equals negative 2.5 centimeters per second

The car’s average velocity is –2.5 cm/s.