Question

(talking about the apollo spacecraft, ignore air resistance) if the 28900kg Apollo 9 CSM sprays 2kg of urine out with a velocity of 20 m/s

a. what velocity would this give the spacecraft

b. over a period of 24 hours, by what distance would this change Apollo’s path?

Answer 1

According to the momentum conservation law, the amount of momentum carried out by urine:

$$p_u = 2kg\times 20m/s = 40kg\cdot m/s$$

is equal to the change of Apollo's momentum:

$$m_a\Delta v_a = p_u$$

where $m_a = 28900kg$ is the mass of Apollo (neglecting the change of mass due to 2kg urine left), and $\Delta v_a$ is the change of its velocity:

$$\Delta v_a = \dfrac{p_u}{m_a}\\ \Delta v_a = \dfrac{40kg\cdot m/s}{28900kg} \approx 0.00138m/s$$

Over the period $t = 24h = 86400s$ this would change Apollo’s path by the following distance:

$$\Delta d = \Delta v_at\\ \Delta d = 0.0014m/s\cdot 86400s \approx 119m$$

Answer. a) 0.00138m/s, b) 119m.