Question

A rocket that is in deep space and initially at rest relative to an inertial reference frame has a mass of 2.60 ✕ 105 kg, of which 1.92 ✕ 105 kg is fuel. The rocket engine is then fired for 250 s while fuel is consumed at the rate of 480 kg/s. The speed of the exhaust product relative to the rocket is 3.15 km/s.

After the 250 s firing, what is the speed of the rocket in m/s?

Sample submission: 1230

Answer #1

Solution- Using the formula

m(t) = m0 - m_dot * t

m(250 s) = (2.60 x 10^5 kg) - (480 kg/s)(250 s) = 1.40 x 10^5 kg

Now final speed

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

As rocket started from rest

Therefore final speed

vf = delta_v

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

m0 = initial mass of rocket = 2.60 x 10^5 kg

m_dot = rate of fuel consumption = 480 kg/s

ve = exhaust speed relative to rocket = 3.15 x 10^3 m/s

T = thrust

T = m_dot * ve = (480 kg/s)(3.15 x 10^3 m/s) = 1.51 x 10^6 N

vf = (T / m_dot) * ln(m0/m) = [1.51 x 10^6 N / (480 kg/s)] * ln(2.60 x 10^5 kg / 1.40 x 10^5 kg)

vf = 1947.39 m/s

A rocket in deep space has an exhaust-gas speed of 2000 m/s.
When the rocket is fully loaded, the mass of the fuel is five times
the mass of the empty rocket. What is the rocket's speed when half
the fuel has been burned?

A model rocket engine has an average thrust of 5.26 N. It has an
initial mass of 25.7 g, which includes fuel mass of 13.0 g. The
duration of its burn is 1.90 s. (Assume all the fuel is consumed.)
(a) What is the average exhaust speed of the engine?
(b) This engine is placed in a rocket body of mass 69.5 g. What
is the magnitude of the final velocity of the rocket if it were to
be fired...

A rocket moving in outer space maintains a constant acceleration
(magnitude = 20 m/s2) while ejecting fuel at aspeed of 15 km/s
relative to the rocket. If the initial mass of the rocket is 3 000
kg, what is the magnitude of thethrust after 800 kg of fuel have
been consumed?

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