As an object (our hailstone) moves it has a change in position from moment to moment. The new position that the object has at the
end of a time interval is the sum of the position that it had at the beginning of the time interval and the change in position. This
change in position, which is also called the displacement , can be approximated by multiplying the rate of change of the position by
the time interval. If the rate of change of position is constant , this product will exactly determine the change in position. By
definition, the rate of change of the position is the velocity . This relationship for the new position can be expressed symbolically as
follows:
x1 = x0 + v * (t1 - t0),
where x1 is the new position, x0 is the old position, v is the velocity, and (t1 - t0) is the time interval. The STELLA modeling
program used in this module will create similar equations.
In the time-based program STELLA, the time interval is referred to as the time step or DT, and represents the time that passes for
each iteration of the modeling scheme. For example, a model has time in units of seconds and the DT is 0.25, then the computer
will calculate a value for each quantity four times for each second of time passing in the model. Note: In STELLA at the equation
level, dt is used. In the run specs, DT is used.
A general and important concept for motion and also time-based computational modeling is this:
What you have = what you had + the change
If the object that is moving is also accelerating , a similar relationship using this concept can be written that relates the velocity to
the acceleration. The new velocity that the object has at the end of a time interval is the sum of the velocity that it had at the
beginning of the time interval and the change in velocity. The change in velocity can be approximated by multiplying the rate of
change of the velocity by the time interval. If the rate of change of velocity is constant , this product will exactly determine the
change in velocity. By definition, the rate of change of the velocity is the acceleration. This can be expressed symbolically as
follows:
v1 = v0 + a * (t1 - t0),
where v1 is the new velocity, v0 is the old velocity, a is the acceleration, and (t1 - t0) is the time interval. If the acceleration is not constant, the motion equations will only give approximate solutions for the new values. However, for a very small unit of time, the acceleration can be thought of as constant, and good approximations can be made using the above equations.
Newton's First Law of Motion , which can be stated in a variety of ways, says that an object with no net external force on it will not accelerate, that is it will not have a change in velocity. Objects seem to resist changes in motion and this resistance is called inertia. The more mass an object has, the more it seems to resist changes in its motion. So objects with more mass have more inertia. Since a net external force is needed to have acceleration, it is important to know what net force means.
Newton's Second Law of Motion describes how much acceleration an object will have based on how much net force is applied to the object and also how much the object resists being accelerated, based on its mass. This can be represented by the relation:
a = Fnet / m
where a is acceleration of the object, Fnet is the net external force on the object, and m is the mass of the object. The acceleration and the net force are always in the same direction.
This physics module focuses on the application of Newton's Second Law of Motion to the hailstone to determine its acceleration from moment to moment. The acceleration is used to then model the motion, based on the motion concepts described above. Accounting for the air resistance force as the hailstone falls is necessary, but the air resistance does not necessarily remain constant. However the air resistance can be calculated at an instant based on the velocity of the hailstone relative to the air, a drag coefficient , the density of the air through which it moves, and the cross-sectional area of the hailstone. The drag coefficient is dependent on a variety of factors . One of these factors is the velocity.
Energy is the ability to do work and create change. In the case of hailstones, the change we are interested in is the damage done as hail impacts other objects. The damage done by the hailstones depends on what it strikes as well as other factors. However, we represent the potential for a hailstone to do damage by the kinetic energy it has at the moment of impact.
This module includes a complex STELLA model that will determine the position, velocity, and acceleration of the hailstone as it falls. The size of the hailstone , its density, and the height from which it falls can be changed. Students can run this model and compare the results to those from the models that they build. The complex STELLA model is available as a java applet. STELLA was chosen as the modeling tool because of the capability to numerically integrate over time. Although other computational tools such as Mathematica or a spreadsheet program like Excel can also accomplish this task, STELLA has some advantages. With a spreadsheet, a small time step for a much larger period of time will result in many rows of data in the spreadsheet. This can be cumbersome, especially in terms of setting up the spreadsheet and also changing the time step as the model is tested and used to generate output. Also, setting up the equations to accurately perform the numerical integration is more difficult. STELLA sets up this part automatically. Mathematica requires significantly more time to learn to set up this type of calculation scheme than STELLA. When students actually build their own models, STELLA has a large time advantage for the novice user. STELLA allows the user to visually set up a graphic organizer that shows how quantities are related. This graphic organizer or map is then used to help create the equations. This visual and organizational feature was considered to be important for the hailstone problem.
gravity_field_ = 9.8
radius_of_hailstone_ = .045
volume_of_hailstone = radius_of_hailstone_^3*4*PI/3
density_of_hailstone = 1000
mass_of_hailstone_ = volume_of_hailstone*density_of_hailstone
weight = -gravity_field_*mass_of_hailstone_
drag_coefficient = .45
INIT velocity = 0
cross_sectional__area = radius_of_hailstone_^2*PI
air_density = 1.29
drag_force = -1*drag_coefficient*velocity*ABS(velocity)*cross_sectional__area*air_density/2
net_force = weight+drag_force
acceleration = net_force/mass_of_hailstone_
rate_of_change_of_position = velocity
initial_height = 1000
INIT position = initial_height
rate_of_change__of_velocity____ = acceleration
kinetic_energy = 0.5*mass_of_hailstone_*velocity^2
air_viscosity = 1.80e-5
Reynolds_Number = 2*radius_of_hailstone_*air_density*ABS(velocity)/air_viscosity
{ RUNTIME EQUATIONS }
velocity(t) = velocity(t - dt) + (rate_of_change__of_velocity____) * dt
position(t) = position(t - dt) + (rate_of_change_of_position) * dt
volume_of_hailstone = radius_of_hailstone_^3*4*PI/3
mass_of_hailstone_ = volume_of_hailstone*density_of_hailstone
weight = -gravity_field_*mass_of_hailstone_
cross_sectional__area = radius_of_hailstone_^2*PI
drag_force = -1*drag_coefficient*velocity*ABS(velocity)*cross_sectional__area*air_density/2
net_force = weight+drag_force
acceleration = net_force/mass_of_hailstone_
rate_of_change_of_position = velocity
rate_of_change__of_velocity____ = acceleration
kinetic_energy = 0.5*mass_of_hailstone_*velocity^2
Reynolds_Number = 2*radius_of_hailstone_*air_density*ABS(velocity)/air_viscosity_
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Last updated 18 July 2002