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Having done that, you can rotate the axis by the same amount in the opposite direction at the end to cause the final solution to apply to the original axes.I will present an example of this in the next section.
Several examples
I will use the information in Figure 1 to analyze several scenarios involving collisions in both one dimension and two dimensions in this section.
The use of JavaScript
All of these examples could be solved using the Google calculator. However, several steps are involved and I find it easier to keep things organized andperform the steps in the correct order by using JavaScript to compute and display the solution.
Note, however, that JavaScript will only do the arithmetic for you. You must still do the algebra/trigonometry yourself. In these examples, I willusually work through the algebra in comment sections and switch to actual code when it is time to compute and display one or more values.
The first one-dimensional scenario involves an automobile accident.
The description as well as the solution to the problem are shown in Listing 1 .
| Listing 1 . The rear end car crash. |
|---|
<!---------------- File JavaScript01.html ---------------------><html><body><script language="JavaScript1.3">/*
This script simulates car #2 rear-ending car #1 in aone-dimensional elastic collision while car #1 was stopped.
The script computes and displays the speed of car #2immediately before the collision under the assumption that
car #1 was moving at 20 m/s just 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*v2yUsing conservation of kinetic energy for the elastic
case gives us one additional equation, allowing usto solve for three unknowns.
0.5*m1*u1^2 + 0.5*m2*u2^2 = 0.5*m1*v1^2 + 0.5*m2*v2^2Variables:
m1, m2, u1, u2, v1, v2, a1, a2, b1, b2*/
document.write("Start Script</br>");
//Solve for u2 and v2 for an elastic collisionvar m1 = 1500;//kg
var m2 = 2000;//kg//Velocities before the collision
var u1 = 0;//meters per second - standing stillvar u2;//unknown value to be determined
//Velocities after the collisionvar v1 = 20;//meters/second
var v2;//unknown value to be determined//Angles
var a1 = 0;//car was not movingvar a2 = 0;//moving straight ahead
var b1 = 0;//moving straight aheadvar b2 = 0;//moving straight ahead
//Convert angles to radiansA1 = a1*Math.PI/180;
A2 = a2*Math.PI/180;B1 = b1*Math.PI/180;
B2 = b2*Math.PI/180;//Compute and print the x and y components of velocity
u1x = u1*Math.cos(A1)u1y = u1*Math.sin(A1)
//u2x = u2*Math.cos(A2)//unknown//u2y = u2*Math.sin(A2)//unknown
v1x = v1*Math.cos(B1)v1y = v1*Math.sin(B1)
//v2x = v2*Math.cos(B2)//unknown//v2y = v2*Math.sin(B2)//unknown
document.write("x and y components of velocity</br>");
document.write("u1x = " + u1x.toFixed(3) + "</br>");
document.write("u1y = " + u1y.toFixed(3) + "</br>");
document.write("v1x = " + v1x.toFixed(3) + "</br>");
document.write("v1y = " + v1y.toFixed(3) + "</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
Eliminate all of the components for which the above printoutshows zero or for which the given values show zero.
0 + m2*u2x = m1*v1x + m2*v2x0 + m2*u2y = 0 + m2*v2y
0 + 0.5*m2*u2^2 = 0.5*m1*v1^2 + 0.5*m2*v2^2Although it isn't totally obvious from the equations, at this
point we need to recognize that because all velocities aredefined to occur along the x-axis, all of the terms in the
middle equation above that deals with the y-component ofvelocity must be zero. Therefore, we can eliminate that
equation entirely.We also need to recognize that because there are no velocity
components along the y-axis, the velocity components alongthe x-axis are actually the magnitudes of those velocity
components. Thus, u2x = u2.Now we will make the substitutions and eliminate terms with a
value of 0 in the process, yieldingm2*u2 = m1*v1 + m2*v2
0.5*m2*u2^2 = 0.5*m1*v1^2 + 0.5*m2*v2^2Substituting known values into the two equations yields
2000*u2 = 1500*20 + 2000*v22000*u2^2 = 1500*20*20 + 2000*v2^2
Simplifying the two equations yieldsu2 = 15 + v2
u2*u2 = 300 + v2*v2Now we need to eliminate one equation through substitution
v2 = u2 - 15u2*u2 = 300 + (u2 - 15)*(u2 - 15)
u2*u2 = 300 + u2*u2 - 30*u2 +225u2*u2 - 300 - u2*u2 + 30*u2 -225 = 0
u2*u2 - u2*u2 + 30*u2 - 300 -225 = 030*u2 - 525 = 0
30*u2 = 525*/
//Compute and print the first speed valuedocument.write("Speed values</br>");
u2 = 525/30;document.write("u2 = " + u2.toFixed(2) + " m/s</br>");
/*Substituting this value back into an earlier energy equation
yieldsv2*v2 = u2*u2 - 300;
*///Compute and display the second speed value
v2 = Math.sqrt(u2*u2 - 300);document.write("v2 = " + v2.toFixed(2) + " m/s</br>");
document.write("==============================="+ "</br>");
//Check the answers for conservation of momentumdocument.write("Check for conservation of momentum</br>");
var mou = m1*u1 + m2*u2;var mov = m1*v1 + m2*v2;
document.write("mou = " + mou.toFixed(0) + " Kg*m/s</br>");
document.write("mov = " + mov.toFixed(0) + " Kg*m/s</br>");
document.write("==============================="+ "</br>");
//Check the answer for elastic collisionvar keIn = 0.5*m1*u1*u1 + 0.5*m2*u2*u2;
var keOut = 0.5*m1*v1*v1 + 0.5*m2*v2*v2;document.write("Check for conservation of energy</br>");
document.write("keIn = " + keIn.toFixed(0)+ " Kg*m^2/s^2</br>");
document.write("keOut = " + keOut.toFixed(0)+ " Kg*m^2/s^2</br>");
document.write("End Script");</script></body></html> |
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