Systems with Varying Mass: A Rocket — Question 2

Question 2

Mass ratio grows exponentially with required velocity change

An ideal rocket in force-free space has constant relative exhaust speed $u=2000\,\mathrm{m/s}$. Its final mass after a burn is $500\,\mathrm{kg}$. It must gain $1000\,\mathrm{m/s}$ along a fixed direction.

1. Find its initial mass and expelled propellant mass.

2. If the required gain doubles, does the propellant mass simply double?

Question 2 - Solution

Mass ratio grows exponentially with required velocity change

(a) Derivation and calculation

Integrate the differential momentum equation: $$dv=-u\frac{dm}{m}\Rightarrow\Delta v=u\ln\frac{m_i}{m_f}.$$ $$\boxed{m_i=m_fe^{\Delta v/u}=500e^{0.5}=824.36\ \mathrm{kg}},$$ $$\boxed{m_p=m_i-m_f=324.36\ \mathrm{kg}}.$$

(b) Result and interpretation

Doubling the required velocity gain gives $$m_p'=500(e^{2000/2000}-1)=\boxed{859.14\ \mathrm{kg}}.$$ $$m_p'>2m_p=648.72\ \mathrm{kg}.$$ The required mass ratio grows exponentially with velocity gain, so propellant does not scale linearly.

Physical check

The final 500 kg must be accelerated throughout the burn.