Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2016

(a) A rocket, moving non-relativistically, has speed v(t)v(t) and mass m(t)m(t) at a time tt after it was fired. It ejects mass with constant speed uu relative to the rocket. Let the total momentum, at time tt, of the system (rocket and ejected mass) in the direction of the motion of the rocket be P(t)P(t). Explain carefully why P(t)P(t) can be written in the form

P(t)=m(t)v(t)0t(v(τ)u)dm(τ)dτdτ(*)\tag{*} P(t)=m(t) v(t)-\int_{0}^{t}(v(\tau)-u) \frac{d m(\tau)}{d \tau} d \tau

If the rocket experiences no external force, show that

mdvdt+udmdt=0(†)\tag{†} m \frac{d v}{d t}+u \frac{d m}{d t}=0

Derive the expression corresponding to ()(*) for the total kinetic energy of the system at time tt. Show that kinetic energy is not necessarily conserved.

(b) Explain carefully how ()(*) should be modified for a rocket moving relativistically, given that there are no external forces. Deduce that

d(mγv)dt=(vu1uv/c2)d(mγ)dt\frac{d(m \gamma v)}{d t}=\left(\frac{v-u}{1-u v / c^{2}}\right) \frac{d(m \gamma)}{d t}

where γ=(1v2/c2)12\gamma=\left(1-v^{2} / c^{2}\right)^{-\frac{1}{2}} and hence that

mγ2dvdt+udmdt=0(‡)\tag{‡} m \gamma^{2} \frac{d v}{d t}+u \frac{d m}{d t}=0

(c) Show that ()(†) and ()(‡) agree in the limit cc \rightarrow \infty. Briefly explain the fact that kinetic energy is not conserved for the non-relativistic rocket, but relativistic energy is conserved for the relativistic rocket.

Typos? Please submit corrections to this page on GitHub.