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: Identify the variables. \begin{itemize} Initial mass of the rocket (\( m_0 \)) = 6000 kg. Rate of mass ejection (\( \dot{m} \)) = 16 kg/s. Relative speed of ejected gases (\( v_r \)) = 11 km/s = 11000 m/s. Time (\( t \)) = 1 minute = 60 seconds. \end{itemize} Step 2: Calculate the mass of the rocket after 1 minute. The mass decreases at a constant rate: \[ m = m_0 - (\dot{m} \cdot t). \] Substitute the values: \[ m = 6000 - (16 \cdot 60) = 6000 - 960 = 5040 \, \mathrm{kg}. \] Step 3: Use the rocket equation to find acceleration. The acceleration of the rocket is given by: \[ a = \frac{\dot{m} \cdot v_r}{m}. \] Substitute the known values: \[ a = \frac{16 \cdot 11000}{5040}. \] Simplify: \[ a = \frac{176000}{5040} = 34.92 \, \mathrm{m/s^2}. \] Thus, the acceleration of the rocket after 1 minute is \( 34.92 \, \mathrm{m/s^2} \).