please help me with question 3 and 4 with full detials(better explain me a few s

Question

please help me with question 3 and 4 with full detials(better explain me a few step please) thank you very much.

The four-force, F mu, is defined in an analogous way as in Newtonian mechanics, F mu = dp mu/d tau where p tau is the relativistic 4-momentum. What physical quantity is given by the zeroth component of this 4-vector? From this result explain why a sub-light particle cannot be accelerated to the speed of light. A spaceship of total mass M 0 (ship, payload, fuel) is initially at rest in space. At some point the ship needs to change speed so that the crew "sees" the destination approaching the ship at speed v s (i.e. the spaceship will travel at speed v s as seen by someone "on the side of the road"). This is done be emitting exhaust from the back of the spaceship. The exhaust is emitted in a very short time period (basically instantly for simplicity). The exhaust speed, as measured in the original frame, is v e. Calculate how much exhaust mass must be emitted in order to achieve the change in speed. (Careful: mass is not necessarily conserved in relativity.)

Explanation / Answer

3)

zeroth component of momentum

P = m c *[1-(v/c)^2]^(-1/2)

[Here i havent used superscripts and also t is same as tau]

F = dP/dt

= m *v/c*[1-(v/c)^2]^(-3/2)

a = F/ m = v/c*[1-(v/c)^2]^(-3/2)

so as v-->c, acceleration a becomes infinity

hence a sublight particle cant be accelerated to velocity of light

4)

let m' be the exhaust mass and m be the remaining mass

applying energy conservation

M_0 c^2 = m' c^2 + mc^2--1

by momentum conservation

m v_s *[1-(v_s/c)^2]^(-1/2)= m' v_e *[1-(v_e/c)^2]^(-1/2)

m = m' (v_e/v_s)* [1-(v_s/c)^2]/[1-(v_e/c)^2]--2

using 1 and 2 we get

m' = M_0 [[1-(v_e/c)^2]]/[(v_e/v_s) + 1]

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