# Solution to determine the velocity of relativistic particles

This solution shows how to develop the algebra to determine the velocity of a nuclear particle fragment resulting from the fragmention of a nuclear particle into two fragments. An example is shown whereby a particle initially at rest fragments into two fragments one of mass (m1) = 1.67 x E-27 Kg. and velocity of (v1) = 0.8c, the other with a mass (m2) = 5.01 x E-27 Kg. The problem asks to determine the unknown velocity (v2) of the second fragment. Special Relativity is considered for this case (since v1~c) and relativistics mechanics and the conservation of linear momentum taken into consideration to determine a general equation to solve for v2.

The particular conditions as set out in the example problem are the used to determine the velocity of the second fragment. The algebra is set out step by step and the logical steps are identified..

#### Solution Preview

Using the conservation of relativistic linear momentum to determine the velocity of a particle mass fragment; a solution developed using simplifying algebraic techniques in a worked example.

PROBLEM:

A particle originally at rest splits into two mass fragments (m1) and (m2), one with a mass of m1 = 1.67 x 10^-27 Kg and a velocity of v1 = 0.8c, the other with a mass of m2 = 5.01 x 10^-27 Kg. Determine the velocity (v2) of the second fragment.

Note: In the text solution that follows, terms such as (v2/c)^2 mean (v2/c) all squared. Please consult the attachment for a clearer understanding where the text may be clearer.

SOLUTION:

STEP 1: Draw a simple diagram to aid understanding -

Figure 1 below shows a visual of the fragmentation process...

{SEE ATTACHMENT FOR ILLUSTRATIVE FIGURE }

Figure 1: Fragmentation process of a particle splitting into two

STEP 2: Formulate the equation describing the conservation of linear ...

#### Solution Summary

This solution shows how one should calculate the velocity of a relativistic particle fragment from the disintergration of a nuclear particle into 2 fragment particles. The solution is developed based on the conservation of linear momentum for the relativistic case and shows how to manipulate the algebra to come up with an expression for the velocity of the particle in question. The problem is then solved on this basis by looking at a certain example with quoted paramters