Question

The engine of a rocket in outer space, far from any planet is turned on. The rocket ejects burnt fuel at constant rate. In the first second of firing, it ejects 1/100 of its initial mass at relative speed of 2000 m/s. The initial acceleration of the rocket is:

• A. \(5 \, \text{m/s}^2\)
• B. \(-10 \, \text{m/s}^2\)
• C. \(+20 \, \text{m/s}^2\)
• D. \(-30 \, \text{m/s}^2\)

Hint:

The rocket thrust equation is a direct application of the conservation of momentum. The force on the rocket is the reaction force to the force required to eject the fuel mass.

Answer 1

The Correct Option is C

Step 1: State the rocket thrust equation. The thrust force \(F\) is calculated as \(F = v_{rel} \left| \frac{dm}{dt} \right|\), where \(v_{rel}\) is exhaust velocity and \(\left| \frac{dm}{dt} \right|\) is the mass ejection rate.
Step 2: Connect thrust to acceleration via Newton's second law. Force \(F\) causes acceleration \(a\) such that \(F = M a\), with \(M\) as the rocket's instantaneous mass. This yields \(M a = v_{rel} \left| \frac{dm}{dt} \right|\), or \(a = \frac{v_{rel}}{M} \left| \frac{dm}{dt} \right|\).
Step 3: Identify parameters for initial acceleration. Use initial mass \(M = M_0\). The relative velocity is \(v_{rel} = 2000 \, \text{m/s}\). The mass ejection rate in the first second is \(\left| \frac{dm}{dt} \right| = \frac{M_0/100}{1 \, \text{s}} = \frac{M_0}{100}\).
Step 4: Compute the initial acceleration. Substitute values into the acceleration formula: \(a_{initial} = \frac{2000 \, \text{m/s}}{M_0} \left( \frac{M_0}{100} \right) = \frac{2000}{100} \, \text{m/s}^2 = 20 \, \text{m/s}^2\). The positive acceleration indicates forward motion.