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: Recall the rocket equation for thrust. The thrust force \(F\) on a rocket is given by the equation: \[ F = v_{rel} \left| \frac{dm}{dt} \right| \] where \(v_{rel}\) is the exhaust velocity relative to the rocket, and \(\left| \frac{dm}{dt} \right|\) is the rate of mass ejection.
Step 2: Relate thrust to acceleration using Newton's second law. The force on the rocket produces an acceleration \(a\) according to \(F = M a\), where \(M\) is the instantaneous mass of the rocket. \[ M a = v_{rel} \left| \frac{dm}{dt} \right| \implies a = \frac{v_{rel}}{M} \left| \frac{dm}{dt} \right| \]
Step 3: Determine the values for the initial acceleration. - We need the initial acceleration, so we use the initial mass of the rocket, \(M = M_0\). - The relative speed is given as \(v_{rel} = 2000 \, \text{m/s}\). - The rocket ejects \(M_0/100\) of its mass in the first second, so the rate of mass ejection is \(\left| \frac{dm}{dt} \right| = \frac{M_0/100}{1 \, \text{s}} = \frac{M_0}{100}\).
Step 4: Calculate the initial acceleration. \[ a_{initial} = \frac{2000 \, \text{m/s}}{M_0} \left( \frac{M_0}{100} \right) = \frac{2000}{100} \, \text{m/s}^2 = 20 \, \text{m/s}^2 \] The acceleration is positive, indicating it is in the forward direction.