A spacecraft of initial mass m_a moves with constant…

A spacecraft of initial mass $m_a$ moves with constant velocity $v_0$. To change its course, a side thruster ejects gas at a constant speed $u$ relative to the craft, perpendicular to the initial velocity, until the mass has dropped to $m_k$. By what angle is the trajectory deflected?

Explanation of Method

Deflecting a spacecraft with a side thruster.

Step 1: Sideways velocity

The thrust acts only sideways, so the rocket equation applies to the sideways component: $v_\perp = u\ln\frac{m_a}{m_k}$.

\[ v_\parallel = v_0 \ \text{(unchanged)}. \]

Step 2: Deflection

\[\theta = \arctan\left(\frac{u}{v_0}\ln\frac{m_a}{m_k}\right)\]

$v_0$ $\Delta v = u\ln\frac{m_a}{m_k}$ $\vec v$ $\theta$

The side thruster adds a velocity $\Delta v$ perpendicular to $\vec v_0$ without changing the forward component, so the path turns by $\theta$ with $\tan\theta = \Delta v/v_0$.

Result

\[\boxed{\,\theta = \arctan\left(\frac{u}{v_0}\ln\frac{m_a}{m_k}\right)\,}\]

Remark

Check. The original result is correct, assuming the jet keeps a fixed direction in space, perpendicular to $\vec v_0$. The final speed is $\sqrt{v_0^2 + u^2\ln^2(m_a/m_k)}$.