This activity requires some knowledge of calculus and application of the Binomial Theorem.Simple relativistic mechanics is applied.

The increase in mass is in accord with

Here m(v) is the mass of a body moving at speed v and mo is the body's rest mass.

If the particle is accelerated from rest under the action of a constant force in Einstein's model we can apply our usual definition of force as rate of change of momentum but obviously mass will change as well as speed as the object is accelerated.
 

Q1) (a) If, as usual F dt = d (mv), then prove that if the force is given by F = km_ocwhere k is a constant, that velocity and time elapsed are related by

(b) Hence deduce that the speed of the particle approaches but never exceeds the speed of light.

Q2) (a) Prove that the distance (s) travelled after time (t) is given by the integral

(b) Hence show that the total distance travelled under the action of this force is given by the equation
 

(c) Deduce that as t gets large then s --> ct

(d) Deduce that if k and t are small then the value of s given by Einstein's model is about ck³t⁴ / 8 LESS than the value of s given by a Newtonian analysis.

(Hint: Use the binomial expansion here)
 

Q3) (a) By eliminating t from previous results prove that
 

(b) Deduce that the increase in mass, m(v) - m_o is proportional to the distance travelled (s) and show that the constant of proportionality is given by m_ok/c