ACCELERATION OF A ROCKET
The engine of a rocket in outer space, far from any planet, is turned on. The rocket ejects burned fuel at a constant rate; in the first second of firing, it ejects \frac{1}{120} of its initial mass m_0 at a relative speed of 2400 m/s. What is the rocket’s initial acceleration?
IDENTIFY and SET UP:
We are given the rocket’s exhaust speed v_{ex} and the fraction of the initial mass lost during the first second of firing, from which we can find dm/dt. We’ll use Eq. (8.39) (a=\frac{d v}{d t}=-\frac{v_{\mathrm{ex}}}{m} \frac{d m}{d t}) to find the acceleration of the rocket.
EXECUTE:
The initial rate of change of mass is
\frac{d m}{d t}=-\frac{m_{0} / 120}{1 \mathrm{~s}}=-\frac{m_{0}}{120 \mathrm{~s}}From Eq. (8.39) (a=\frac{d v}{d t}=-\frac{v_{\mathrm{ex}}}{m} \frac{d m}{d t}),
a=-\frac{v_{\mathrm{ex}}}{m_{0}} \frac{d m}{d t}=-\frac{2400 \mathrm{~m} / \mathrm{s}}{m_{0}}\left(-\frac{m_{0}}{120 \mathrm{~s}}\right)=20 \mathrm{~m} / \mathrm{s}^{2}
EVALUATE: The answer doesn’t depend on m_0. If v_{ex} is the same, the initial acceleration is the same for a 120,000-kg spacecraft that ejects 1000 kg/s as for a 60-kg astronaut equipped with a small rocket that ejects 0.5 kg/s.