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Fysik B You know you want it!
http://en.wikipedia.org/wiki/Elastic_collision
Equations
One-dimensional Newtonian
Consider two particles, denoted by subscripts 1 and 2. Let m be the mass, u be the velocity before collision and v be the velocity after collision.
Total kinetic energy is the same before and after the collision, hence:
Total momentum remains constant throughout the collision:
These equations may be solved directly to find and . However, the algebra can get messy. A cleaner solution is to first change the frame of reference such that either or appears to be 0. The final velocities in the new frame of reference canthen be determined followed by a conversion back to the original frameof reference to reach the same final result. Once either or is determined the other may be found by symmetry.
Solving these simultaneous equations we get:
,
OR
, .
The latter is the trivial solution, corresponding to the case that no collision has taken place (yet).
For example:
Ball 1: mass = 3 kg, v = 4 m/sBall 2: mass = 5 kg, v = −6 m/s
After collision:
Ball 1: v = −8.5 m/sBall 2: v = 1.5 m/s
Property:
Derivation: Using the kinetic energy we can write
Rearrange momentum equation:
Dividing kinetic energy equation by the momentum equation we get:
- the relative velocity of one particle with respect to the other is reversed by the collision
- the average of the momenta before and after the collision is the same for both particles
Elastic collision of equal masses
As can be expected, the solution is invariant under adding aconstant to all velocities, which is like using a frame of referencewith constant translational velocity.
Elastic collision of masses in a system with a moving frame of reference
The velocity of the center of mass does not change by the collision:
The center of mass at time before the collision and at time after the collision is given by two equations:
, and
Hence, the velocities of the center of mass before and after the collision are:
, and
The numerator of is the total momentum before the collsion, and numerator of is the total momentum after the collsion. Since momentum is conserved, we have .
With respect to the center of mass both velocities are reversed bythe collision: in the case of particles of different mass, a heavyparticle moves slowly toward the center of mass, and bounces back withthe same low speed, and a light particle moves fast toward the centerof mass, and bounces back with the same high speed.
From the equations for and above we see that in the case of a large , the value of is small if the masses are approximately the same: hitting a muchlighter particle does not change the velocity much, hitting a muchheavier particle causes the fast particle to bounce back with highspeed.
Elastic collision of unequal masses
Therefore a neutron moderator (a medium which slows down fast neutrons, thereby turning them into thermal neutrons capable of sustaining a chain reaction)is a material full of atoms with light nuclei (with the additionalproperty that they do not easily absorb neutrons): the lightest nucleihave about the same mass as a neutron.
One-dimensional relativistic
According to Special Relativity,
Where p denotes momentum of any massive particle, v denotes velocity, c denotes the speed of light.
in the center of momentum frame where the total momentum equals zero,
p1 = − p2u1 = − v1
Where m1 represents the rest mass of the first colliding body, m2 represents the rest mass of the second colliding body, u1 represents the initial velocity of the first collidng body, u2 represents the initial velocity of the second colliding body, v1 represents the velocity after collision of the first colliding body, v2 represents the velocity after collision of the second colliding body, p1 denotes the momentum of the first colliding body, p2 denotes the momentum of the second colliding body and c denotes the speed of light in vacuum, E denotes the total energy of the system (i.e. the sum of rest masses and kinetic energies of the colliding bodies).
Since the total energy and momentum of the system are conserved andthe rest mass of the colliding body do not change, it is shown that themomentum of the colliding body is decided by the rest masses of thecolliding bodies, total energy and the total momentum. The magnitude ofthe momentum of the colliding body does not change after collision butthe direction of movement is opposite relative to the center of momentum frame.
Classical Mechanics is only a good approximation. It will give accurate results when itdeals with the object which is macroscopic and running with much lowerspeed than the speed of light.Beyond the classical limits, it will give a wrong result. Totalmomentum of the two colliding bodies is frame-dependent. In the center of momentum frame c, according to Classical Mechanics,
It is shown that u1 = − v1 remains true in relativistic calculation despite other differences. Oneof the postulates in Special Relativity states that the Laws of Physicsshould be invariant in all inertial frames of reference. That is, iftotal momentum is conserved in a particular inertial frame ofreference, total momentum will also be conserved in any inertial frameof reference, although the amount of total momentum is frame-dependent.Therefore, by transforming from an inertial frame of reference toanother, we will be able to get the desired results. In a particularframe of reference where the total momentum could be any,
We can look at the two moving bodies as one system of which the total momentum is pT, the total energy is E and its velocity vc is the velocity of its center of mass. Relative to the center ofmomentum frame the total momentum equals zero. It can be shown that vc is given by:
Now the velocities before the collision in the center of momentum frame u1' and u2' are:
v1' = − u1'v2' = − u2'
When u1 < < c and u2 < < c,
pT ≈ m1u1 + m2u2vc ≈ u1' ≈ u1 − vc ≈ u2' ≈ v1' ≈ v2' ≈ v1 ≈ v1' + vc ≈ v2 ≈
Therefore, the classical calculation only holds true when the speedof both colliding bodies is much lower than the speed of light (about300 million m/s).
http://en.wikipedia.org/wiki/Elastic_collision
Equations
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