Question

A research space vehicle in gravity-free and drag-free outer space launches a smaller spacecraft into a meteor shower region. The 2-kg sensitive instrument package of this spacecraft (25 kg total mass) limits the maximum acceleration to no more than 50 m/sec2. It is launched by a solid propellant rocket motor (Is = 260 sec and ζ = 0.88). Assume instant start and stop of rocket motor.

(a) Determine the maximum allowable burn time, assuming steady constant propellant mass flow;

Answer 1

The maximum allowable burn time is calculated using the maximum thrust determined by the sensitive instrument's acceleration limit. This is related to the mass flow rate and specific impulse of the rocket motor. By isolating the mass flow rate, the burn time can be found based on the total consumed propellant mass.

Step-by-step explanation:

To determine the maximum allowable burn time for the rocket motor, we must calculate it based on the limitations of the sensitive instrument's maximum acceleration. The rocket motor has a specific impulse (Is) of 260 seconds and an efficiency (ζ) of 0.88. The thrust (F) produced by the rocket motor can be calculated using the product of the mass flow rate (ṁ), the gravitational acceleration (g), and Is. The maximum force due to F is equal to the product of the mass of the instrument package (m) and the maximum acceleration (a). So, F = ma. Now we have F = 2 kg * 50 m/s2 = 100 N. To find ṁ, we use F = ṁ * g * Is * ζ. Isolate the mass flow rate to get ṁ = F / (g * Is * ζ). Plugging the numbers in yields the mass flow rate.

The burn time (t) can then be estimated based on the total mass of propellant consumed divided by ṁ, considering the rocket motor starts and stops instantaneously. As the mass of propellant is a portion of the total spacecraft mass (25kg), excluding the instrument package (2kg), we consider the propellant mass to be a fraction of the remaining 23kg.