Question

A rocket of initial mass 6000 kg ejects gases at a constant rate of 16 kg/s with a constant relative speed of 11 km/s. What is the acceleration of the rocket one minute after the blast?

Hint:

The acceleration of a rocket can be calculated using the formula \( a = \frac{\dot{m} \cdot v_r}{m} \), where \( \dot{m} \) is the rate of mass ejection, \( v_r \) is the relative speed of ejected gases, and \( m \) is the remaining mass.

Answer 1

Step 1: Variable Identification. \begin{itemize} Initial rocket mass (\( m_0 \)) = 6000 kg. Mass ejection rate (\( \dot{m} \)) = 16 kg/s. Ejected gas relative speed (\( v_r \)) = 11 km/s = 11000 m/s. Duration (\( t \)) = 1 minute = 60 seconds.\end{itemize}Step 2: Rocket mass calculation at 1 minute. Mass change is linear:\[m = m_0 - (\dot{m} \cdot t).\]Substitution:\[m = 6000 - (16 \cdot 60) = 6000 - 960 = 5040 \, \mathrm{kg}.\]Step 3: Acceleration determination using the rocket equation. Rocket acceleration formula:\[a = \frac{\dot{m} \cdot v_r}{m}.\]Value substitution:\[a = \frac{16 \cdot 11000}{5040}.\]Result:\[a = \frac{176000}{5040} = 34.92 \, \mathrm{m/s^2}.\]The rocket's acceleration after 1 minute is \( 34.92 \, \mathrm{m/s^2} \).