What does the model look like?

How do we model a marble rolling down a ramp?

The STELLA model attempts to duplicate the experiment that you just performed. It is built in two sections. In the first we describe the motion of the marble down the inclined plane. In the second we describe the flight  of the marble in the air from the time it leaves the incline until it lands in the container. 

Motion Down The Inclined Plane

The marble on the inclined plane is pulled down the plane by the force of gravity. The component of that force along the plane is mgSIN(ø), where m is the mass of the marble, g is the acceleration due to gravity and ø is the angle the inclined plane makes with the horizontal. Using Newton's Law of Motion; 

            dV/dt = gSIN(ø) 

where dV is the change in speed along the inclined plane, dt is the change in time, and gSIN(ø) represents the acceleration of the marble along the ramp. 

In the STELLA model V is represented by a stock and gSIN(ø) is a flow that derives its values from a gravity rate converter and an incline angle converter. 

The location of the marble on the inclined plane at any time is determined by the definition of speed; 

          dS/dt = V    Where dS is the change in distance from the starting point.

Here is the model of the Rolling Marble for the inclined plane portion of the activity. 

Alternative Assignment

Create a STELLA model for a ball rolling down an inclined plane. Assume the gravity rate, incline length, incline height and initial speed are specified. Calculate the speed of travel and the distance travelled down the incline at every instant of time. 

Motion In The Air

The marble in the air falls under the influence of gravity. Its downward speed, V_y , is given by the formula derived from Newton's second law of Motion; 

          dV_y/dt = g

The marble in the air moves horizontally solely because of its speed as it leaves the incline. Gravity has no effect on the horizontal velocity. The horizontal speed, V_x , is given by; 

         dV_x/dt = 0

The distance travelled in the air is again given by the definitions of speed; 

         dS_y/dt = V_y                    dS_x/dt = V_x

Here is the model of the Rolling Marble for the portion of the activity when the marble is in the air. 

Alternative Assignment

Create a STELLA model for a projectile in space. Assume the gravity rate, the maximum allowable falling distance (distance to ground), initial horizontal speed and initial vertical speed are specified. Calculate the horizontal speed, the vertical speed, the horizontal distance travelled and the vertical distance travelled at every instant of time. Plot the path of the projectile. 

Model of inclined plane combined with projectile motion

The combined model incorporates all of the above along with additional factors, namely; 

• The length of the inclined plane.
• The vertical distance between the two ends of the inclined plane.
• The distance from the bottom of the incline to the floor.

The image at the right (click on the graphic to see full-sized model in a separate window)