Question

Consider a relativistic particle of mass M and kinetic energy K. derive an expression for the particle's speed U in terms of K and M.

show steps please

Answer 1

The famous Einstein relationship for energy

can be blended with the relativistic momentum expression

mov v2

to give an alternative expression for energy.
The combination pc shows up often in relativistic mechanics. It can be manipulated as follows:

0 2 2 c2

and by adding and subtracting a term it can be put in the form:

which may be rearranged to give the expression for energy:

212

Note that the m with the zero subscript is the rest mass, and that m without a subscript is the effective relativistic mass.