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The output
The output produced by this script is shown in Figure 2 .
The only comments that I will make in addition to the comments in Listing 1 are that the values displayed for mou, mov, keIn, and keOut near the end of Figure 2 show that both momentum and kinetic energy were conserved. Therefore, the requirements for an elastic collision were met.
The meaning of each of those terms is as follows:
| Figure 2 . Output for the rear end car crash. |
|---|
Start Script
x and y components of velocityu1x = 0.000
u1y = 0.000v1x = 20.000
v1y = 0.000===============================
Speed valuesu2 = 17.50 m/s
v2 = 2.50 m/s===============================
Check for conservation of momentummou = 35000 Kg*m/s
mov = 35000 Kg*m/s===============================
Check for conservation of energykeIn = 306250 Kg*m^2/s^2
keOut = 306250 Kg*m^2/s^2End Script |
The description and the solution to another rear end car crash are provided in Listing 2 . Whereas the previous car-crash scenario described an elastic collision, this scenario describes a perfectly inelastic collision.In this crash, the two cars become entangled and move forward asa single object following the collision. Therefore, this is an example of a perfectly inelastic collision.
| Listing 2 . A perfectly inelastic car crash. |
|---|
<!---------------- File JavaScript10.html ---------------------><html><body><script language="JavaScript1.3">/*
This script simulates car #2 rear-ending car #1 in aone-dimensional perfectly inelastic collision while car #1 was
stopped.Assume that car #2 was moving at 17.5 m/s immediately before
the collision. The cars became entangled and moved as a singleobject in a straight line following the collision. Find
the velocity of the two objects immediately following thecollision. Find the total momentum of the two objects
immediately before and immediately after the collision.Using conservation of momentum alone, we have two
equations, allowing us to solve for two unknowns.m1*u1x + m2*u2x = m1*v1x + m2*v2x
m1*u1y + m2*u2y = m1*v1y + m2*v2yHowever, because the cars are moving along the x-axis, all
terms in the second equation must be zero. This limits us toonly the first equation shown above. Note also that because
this is an inelastic collision, we can't use the equationbased on conservation of energy.
Variables:m1, m2, u1, u2, v1, v2, a1, a2, b1, b2
*/document.write("Start Script</br>");
var m1 = 1500;//kgvar m2 = 2000;//kg
//Velocities before the collisionvar u1 = 0;//meters per second - standing still
var u2 = 17.5;//meters per second//Velocities after the collision
var v1;//unknown but v2 = v1var v2;//unknown value to be determined
//Anglesvar a1 = 0;//car was not moving
var a2 = 0;//moving straight aheadvar b1;//unknown but not needed
var b2;//unknown but not needed//Convert angles to radians
A1 = a1*Math.PI/180;A2 = a2*Math.PI/180;
//B1 = b1*Math.PI/180;//B2 = b2*Math.PI/180;
//Compute and print the initial x and y components of velocity.// Because both cars are moving along the x-axis, all
// y-components are zero and all x components are equal to the// magnitude.
u1x = u1u1y = 0
u2x = u2u2y = 0
//v1x = v1//unknownv1y = 0
//v2x = v2//unknownv2y = 0
document.write("Initial velocities</br>");
document.write("u1x = " + u1x.toFixed(2) + "</br>");
document.write("u2x = " + u2x.toFixed(2) + "</br>");
document.write("==============================="+ "</br>");
/*Prepare the equations for use in solving the problem.
Given the following three equationsm1*u1x + m2*u2x = m1*v1x + m2*v2x
m1*u1y + m2*u2y = m1*v1y + m2*v2y0.5*m1*u1^2 + 0.5*m2*u2^2 = 0.5*m1*v1^2 + 0.5*m2*v2^2
Only the first equation can be used for an inelastic collisionmoving along the x-axis. This gives us the following equation
to work with.m1*u1x + m2*u2x = m1*v1x + m2*v2x
Eliminate all of the components which are known to be zero.m2*u2x = m1*v1x + m2*v2x
For a perfectly inelastic collision, v2=v1 yieldingm2*u2x = m1*v1x + m2*v1x, or
m2*u2x = (m1 + m2)*v1xRearranging terms gives the following:
*///Compute and print the final speed values
v1x = m2*u2x/(m1 + m2)v2x = v1x;//required for a perfectly inelastic collision
document.write("Final speed values</br>");
document.write("v1x = " + v1x.toFixed(2) + " m/s</br>");
document.write("v2x = " + v2x.toFixed(2) + " m/s</br>");
document.write("==============================="+ "</br>");
//Check the answer for conservation of momentumdocument.write("Check for conservation of momentum</br>");
var mou = m1*u1x + m2*u2x;//momentum before the collisionvar mov = m1*v1x + m2*v2x;//momentum after the collision
document.write("mou = " + mou.toFixed(0) + " Kg*m/s</br>");
document.write("mov = " + mov.toFixed(0) + " Kg*m/s</br>");
document.write("==============================="+ "</br>");
document.write("End Script");</script></body></html> |
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