A rocket is launched in a space habitat, where there is essentially no gravitational force. The rocket is subject to a linear drag force from the habitat's atmosphere, \( \vec{f}=-b v \hat{\boldsymbol{v}} \) with \( b \) a positive constant, \( v=|\vec{v}| \) is the magnitude of the velocity, and \( \hat{\boldsymbol{v}} \) the unit vector in the direction of the velocity. The rocket ejects mass with exhaust velocity \( v_{\text {ex }} \) at a constant rate given by \( \dot{m}=-k \), with \( k \) a positive constant. The rocket starts from rest \( \left(v_{0}=0\right) \), and has initial mass \( m_{0} \). (a) What are the SI units or dimensions of \( b \) ? (b) What are the SI units or dimensions of \( k \) ? (c) Find the terminal velocity of the rocket in terms of the other constants given above, assuming the fuel does not run out before the rocket achieves terminal velocity. (d) Find \( v(m) \).
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