Projectile Motion
This Applet can be used to study the Physics of projectile motion in several ways. First, the applet visualizes the parabolic trajectory of a projectile shot from a cannon. The user can vary Vo, the initial velocity, and Theta, the initial angle to determine their effects on the path. Students can then apply the basic kinematics equations to predict the maximum height and the distance of the trajectory. The height can be determined, after the projectile has landed, by moving the mouse over the point and clicking;the x and y value of the point will be displayed in the top-right portion of the screen. The equations to be employed are:
x = Vo* cos(theta) * time
y= Vo*sin(theta) * time - 1/2 * g * time^2 + yo
Vy = Vo*sin(theta) - g * time.
Vy^2 = (Vo*sin(theta))^2 - 2*g*y
NOTE: The initial x value is 0, but yo is 15.
Students can check their results by switching to the table view and viewing the true x and y values for every time step (depending on dt).
The calculus of projectile motion can be taught by analyzing the graphs available in the graph choice bar. Selecting "X and Vx" , or "Y and Vy" creates a new Frame (make sure that you position both open frames so that both can be viewed simultaneously) with a graph of the x, or y, position and velocity as a function of time. The linear slope of Vy, as the derivative of the parabolic y trajectory can be easily seen. The constancy of Vx as the derivative of the function of x position vs time can be viewed as well.
The energy of the projectile can also be observed by selecting "Energy" from the graph choice bar. This creates a new graph which displays the negative linear relationship between KE and PE. As KE decreases, PE increases and vice-versa. This of course satisfies the conservation of energy law:
E = KE + PE
One of the greatest virtues of this Applet is its ability to visualize the effects of forces that are often neglected, due to their complexity, in introductory physics. By selecting the Resistance box, the relative effects of air resistance are displayed on the screen. Notice that the second (shorter) trajectory is no longer parabolic.
It is important to note that the actual drag force, air resistance, is a complicated function of the velocity of the projectile, the mass of the projectile, the wind velocity, the drag coefficient of the projectile and the density of the air. For this animation, the density of the air was neglected, the mass of the projectile was set to 1.0 kg (for simplicity) and the drag coefficient, B, was set to .003 due to the fact that this value appropriately models a general effect of drag on the projectile. The wind velocity can still be varied, both positively and negatively.
The algorithms employed to calculate the drag forces on the x and y components of the velocity were:
Velocity, V= sqrt( Vx^2 + Vy^2)
Drag Force, x = B* | (V-Vw)*(V-Vx) |
Drag Force, y= B* | (V-Vw)*Vy |
Notice that these resemble the general equation Fdrag = B*v^2 for resistance.
With exactness as an improbability, these values appropriately enough model the effects of real life to suit educational purposes. More exactness would require more variables and make the Applet much less user-friendly. When studying the effects of resistance, make sure to reanalyze the graphs of position and velocity to observe that they are not as simple as before. Also, reanalyze the energy graph to observe that energy is no longer conserved. Instead the conservation of energy must add a much more complicated factor:
E = KE + PE + ò Fdrag ds