## Graphing Calculator Solution for Equation of Motion Problems

Graphing calculators can sometimes be used to solve motion problems that involve two or more objects. Enough information needs to be provided enabling one to express the objects' distance (displacement) as a function of time (t). These equations are then graphed and displayed using a graphing calculator. It is then easy to determine elapsed time, distance (or displacement), and also speed (or velocity) at a certain point.

Example: Car one is at rest at an intersection. At the instant that a truck traveling in the same direction at a constant rate of 12 m/s passes the car, it begins to accelerate at a uniform rate of 2.8 m/s2. How long does it take for the car to overtake the truck? How far has it gone at this point? What is its speed at this point?

For the purposes of the problem, the car and the truck start at the same point. Both travel the same distance, d, in the same amount of time, t(hint; the interpretation of "overtake.").

Mathematically, the distance of an object as a function of time can be expressed as:

d(t) = di + vit + 1/2 at2
Where di is the initial displacement of the object (example: the object's motion started 20 m away from the other object's motion) and vi is the initial speed of the car (If the object is traveling at constant speed, vi=v and a=0).

What do we know about the car?

vi=0
a=2.8 m/s2
d=d
t=t

Write an equation of motion for the car:

d(t) = 0 + 0(t) + 1/2 (2.8)t2
or, d=1.4t2

What do we know about the truck?

vi=v=12 m/s
d=d
t=t

Write an equation of motion for the truck:

d(t) = 0 + 12(t) + 1/2 (0)t2
or, d= 12 t

These equations of motion can now be entered into the graphing calculator and displayed. Your x axis represents time and your y axis represents distance.

On the TI-83, you will be using the five "graphing keys" just below the display. Select y= and enter y1=1.4x2 and y2=12x. Select GRAPH. To get your two graphs to fit the display screen, select WINDOW. Appropriate settings for this example are xmin=0, xmax=10, xscl=1, ymin=0, ymax=150, and yscl=5.

To answer the questions "How long does it take for the car to overtake the truck?" and "How far has it gone at this point?" you need to determine the intersection of your two plots. On the TI-83, select 2nd, CALC. Choose 5: intersect (remember, both objects are in the same place at the same time in this problem). The calculator will ask you about the first curve, the second curve, and a guess. Hit ENTER for each. It will then calcualte the intersection. The x value represents the time to overtake and the value represents the distance traveled.

To answer the question "What is its speed at this point?," select 2nd, CALC. Choose 6: dy/dx. Your cursor should be at the intersection point (or the point where you want to determine the speed) on the curve representing the motion of the accelerating object. Hit ENTER, ENTER. The value displayed at the bottom is the object's speed.