Question

The initial mass of a rocket is 1000 kg. Calculate at what rate the fuel should be burnt so that the rocket is given an acceleration of 20 \(ms^{-2}\). The gases come out at a relative speed of 500 \(ms^{-1}\) with respect to the rocket : [Use \(g = 10 \text{ m/s}^2\)]

• A. 60 kg \(s^{-1}\)
• B. \(6.0 \times 10^2\) kg \(s^{-1}\)
• C. 500 kg \(s^{-1}\)
• D. 10 kg \(s^{-1}\)

Hint:

Don't forget to account for the weight (\(Mg\)) of the rocket. The thrust must lift the rocket first before it can accelerate it upwards.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Rocket propulsion is based on Newton's third law and the conservation of momentum. The thrust force generated by the ejected fuel must overcome gravity and provide the required acceleration.
Step 2: Key Formula or Approach:
The net force equation for the rocket is:
\[ F_{thrust} - Mg = Ma \]
Thrust force \(F_{thrust} = v_{rel} \left( \frac{dm}{dt} \right)\).
Step 3: Detailed Explanation:
Given:
\(M = 1000 \text{ kg}\)
\(a = 20 \text{ m/s}^2\)
\(g = 10 \text{ m/s}^2\)
\(v_{rel} = 500 \text{ m/s}\)
Calculate the required thrust:
\[ F_{thrust} = M(g + a) = 1000(10 + 20) = 1000 \times 30 = 30000 \text{ N} \]
Calculate the fuel burn rate \(\frac{dm}{dt}\):
\[ v_{rel} \left( \frac{dm}{dt} \right) = 30000 \]
\[ \frac{dm}{dt} = \frac{30000}{500} = 60 \text{ kg/s} \]
Step 4: Final Answer:
The fuel should be burnt at a rate of 60 kg \(s^{-1}\).