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I recommend that you also study the other lessons in my extensive collection of online programming tutorials. You will find a consolidated index at www.DickBaldwin.com .
I have touched on collisions in one dimension in earlier modules. I will deal with collisions in a more rigorous manner in this module, and will also extendthe analysis to two dimensions.
Facts worth remembering -- Types of collisions
An elastic collision is one in which the total kinetic energy is the same before and after the collision.
An inelastic collision is one in which the final kinetic energy is less than the initial kinetic energy.
A perfectly inelastic collision is one that results in the two objects sticking together. The decrease of kinetic energy in a perfectly inelastic collision is aslarge as possible (consistent with the conservation of momentum).
Momentum is conserved regardless of whether the collision is elastic or inelastic.
A general solution for elastic collisions
I will provide you with three equations that apply in general to elastic collisions in two dimensions. However, as you will see, there are more thanthree variables involved in such collisions. With only three equations, you can only solve for three unknowns. Therefore, in order to solve the general problem,the values of all the other variables must be known.
The two-dimensional solution can be applied to one-dimensional problems by constraining the directions of motion of the two objects to either be the sameor to differ by 180 degrees. If possible, such problems should be structured to cause the directions to be along the x-axis. This will often simplify thesolution.
A general solution for inelastic collisions
The case for inelastic collisions is more restrictive than the case for elastic collisions. Only two of theequations mentioned above apply to inelastic collisions. As a result, you can only solve for two unknown values for an inelastic collision. The values for allof the other variables must be known.
Collision equations
The equations for collisions of two objects in two-dimensional space are shown in Figure 1 . Note that it is assumed that the two objects constitute an isolated system -- that is, a closed system that is not subject to externalinteractions. This requires that both the magnitude and the direction of the momentum for the system be the same at the beginning and the end of the collision.
Figure 1 . Equations for collisions of two objects in two-dimensional space. |
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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 elasticcase gives us one additional equation, allowing us
to solve for three unknowns.0.5*m1*u1^2 + 0.5*m2*u2^2 = 0.5*m1*v1^2 + 0.5*m2*v2^2
Velocities can be decomposed into their x andy-components using the following equations:
u1x = u1*cos(a1)u1y = u1*sin(a1)
u2x = u2*cos(a2)u2y = u2*sin(a2)
v1x = v1*cos(b1)v1y = v1*sin(b1)
v2x = v2*cos(b2)v2y = v2*sin(b2)
Substitution yields the following for the two momentumequations:
m1*u1*cos(a1) + m2*u2*cos(a2) = m1*v1*cos(b1) + m2*v2*cos(b2)m1*u1*sin(a1) + m2*u2*sin(a2) = m1*v1*sin(b1) + m2*v2*sin(b2)
where:m1 and m2 are the masses of the two objects in kg
u1 and u2 are the magnitudes of the initialvelocities of the two objects. Velocities are
measured in meters/secondv1 and v2 are the magnitudes of the final velocities
of the two objectsu1x, u1y, v1x, and v1y are the x and y-components
of the initial and final velocities in 2D space.a1 and a2 are angles that describe the initial
directions of the two objects through 2D space.Angles are measured counter-clockwise relative to
the positive x-axisb1 and b2 are angles that describe the final
directions of the two objects through 2D spaceIt is assumed that the two objects constitute an
isolated system.Variables:
m1, m2, u1, u2, v1, v2, a1, a2, b1, b2 |
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