Question

A 5000 kg rocket is set for vertical firing. The exhaust speed is 800 m/s. To give an initial upward acceleration of 20$m/s^{2},$ the amount of gas ejected per second to supply the needed thrust will be (Take g = 10 $m/s^{2}$)

• A. 127.5 kg/s
• B. 137.5 kg/s
• C. 155.5 kg/s
• D. 187.5 kg/s

Answer 1

The Correct Option is D

 To solve the problem of calculating the amount of gas ejected per second to give the rocket its initial upward acceleration, we apply the principles of classical mechanics. We begin by considering the forces acting on the rocket and the thrust formula derived from Newton's second law.

Step-by-Step Solution:

1. Identify the total force required for the given acceleration:
2. The rocket needs an initial upward acceleration \( a = 20 \, \text{m/s}^2 \). The gravitational force is \( mg = 5000 \times 10 = 50000 \, \text{N} \), where \( m = 5000 \, \text{kg} \) and \( g = 10 \, \text{m/s}^2 \).
3. The total force \( F \) needed to achieve this net acceleration is given by:
4. Substituting the known values:
5. Thrust provided by the ejected gas is given by the formula:
6. Substitute the thrust \( F = 150000 \, \text{N} \) and exhaust speed \(\text{Ve} = 800 \, \text{m/s}\):
7. Solve for \(\frac{\Delta m}{\Delta t}\):

Therefore, the mass of the gas ejected per second to provide the required thrust and acceleration is 187.5 kg/s.

Conclusion:

The correct answer is the option 187.5 kg/s.