This example requires an understanding of the relationships between acceleration and velocity of an object moving in a straight line. A clear discussion of this relationship can be found in "Acceleration" (http://cnx.org/content/m13769/latest/); the Wikipedia article "Motion Graphs and Derivatives" (http://en.wikipedia.org/wiki/Motion_graphs_and_derivatives) also has an explanation of this relationship as well as a discussion of average and instantaneous velocity and acceleration and the role derivatives play. Also, in this example, we will compute approximate integrals using the trapezoidal rule; the Wikipedia article "Trapezium rule" (http://en.wikipedia.org/wiki/Trapezoidal_rule) has an explanation of the trapezoidal rule.
Velocity Analysis of an Experimental Rail Gun
A railgun is a device that uses electrical energy to accelerate a projectile; information about railguns can be found at the Wikipedia article "Railgun" (http://en.wikipedia.org/wiki/Railgun). The paper "Effect of Railgun Electrodynamics on Projectile Launch Dynamics" by Zielinski shows the current profile of a railgun launch. The acelleration a of the projectile (in units of ms2 ) is a function of the current c through the projectile (in units of kAmp). This function is given by the equation
where sgn(c) is 1 if c>0 and −1 if c<0.
Download the data set of current values in the file Current.txt (available at http://cnx.org/content/m14031/latest/Current.txt) onto your computer. The file is formatted as two columns: the first column is time in miliseconds, and the second column is current in kA.
The following sequence of commands will load the data, create a vector
of time values, create a vector
of current values, and plot the current as a function of time.
load Current.txt -ascii
t = Current(:,1);
c = Current(:,2);
The plot should be similar to that in Figure 6.
Plot of railgun current versus time
Compute the projectile velocity as a function of time. Note that velocity is the integral of acceleration.