Physics Problem

Question

A rocket, which is in deep space and initially at rest relative to an inertial reference frame, has a mass of 88.3 × 105 kg, of which 17.7 × 105 kg is fuel. The rocket engine is then fired for 450 s, during which fuel is consumed at the rate of 330 kg/s. The speed of the exhaust products relative to the rocket is 3.24 km/s. (a) What is the rocket's thrust? After the 450 s firing, what are (b) the mass and (c) the speed of the rocket?

Answer 1

m0 = initial mass of rocket = 88.3 x 10^5 kg

 

m_dot = rate of fuel consumption = 330 kg/s

 

ve = exhaust speed relative to rocket = 3.24 x 10^3 m/s

 

(a)

 

T = thrust

 

T = m_dot * ve = (330 kg/s)(3.24 x 10^3 m/s) = 1.0692 x 10^6 N

 

(b)

 

m(t) = m0 - m_dot * t

 

m(450 s) = (88.3 x 10^5 kg)  - (330 kg/s)(450 s) = 86.815 x 10^5 kg

 

For final speed, use:

 

delta_v = g * Isp * ln(m0/m), where Isp = specific impulse = T / (m_dot * g)

 

Since the rocket started from rest, the final speed vf = delta_v

 

vf = g * [T / (m_dot * g)] * ln(m0/m)

 

vf = (T / m_dot) * ln(m0/m) = [1.0692 x 10^6 N / (330 kg/s)] * ln(88.3 x 10^5 kg /  86.815 x 10^5 kg)

 

vf = 54.95 m/s