Q 84

Expert-verifiedFound in: Page 291

Book edition
4th

Author(s)
Randall D. Knight

Pages
1240 pages

ISBN
9780133942651

Section 11.6 found an equation for v_{max} of a rocket fired in deep space. What is v_{max} for a rocket fired vertically from the surface of an airless planet with free-fall acceleration g? Referring to Section 11.6, you can write an equation for ∆P_{y}, the change of momentum in the vertical direction, in terms of dm and dv_{y}. ∆P_{y} is no longer zero because now gravity delivers an impulse. Rewrite the momentum equation by including the impulse due to gravity during the time dt during which the mass changes by dm. Pay attention to signs! Your equation will have three differentials, but two are related through the fuel burn rate R. Use this relationship—again pay attention to signs; m is decreasing—to write your equation in terms of dm and dv_{y}. Then integrate to find a modified expression for v_{max} at the instant all the fuel has been burned.

a. What is v_{max} for a vertical launch from an airless planet ? Your answer will be in terms of m_{R}, the empty rocket mass; m_{F0}, the initial fuel mass; v_{ex}, the exhaust speed; R, the fuel burn rate; and g.

b. A rocket with a total mass of 330,000 kg when fully loaded burns all 280,000 kg of fuel in 250 s. The engines generate 4.1 MN of thrust. What is this rocket’s speed at the instant all the fuel has been burned if it is launched in deep space ? If it is launched vertically from the earth?

We need to find out :

a) v_{max} for a vertical launch from an airless planet

b) Rocket’s speed at the instant all the fuel has been burned if it is launched in deep space

Now dividing equation1 by dt both sides,

Mass of rocket with fuel, m_{o} = 330,000 kg

Mass of fuel, m_{f} = 280,000 kg

Mass of rocket, m_{r} = 330,000 - 280,000 = 50,000 kg

Rocket’s speed at the instant when all the fuel has been burned if it is launched in deep space is 9,910 m/s.

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