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Q 84

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
Found in: Page 291

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Short Answer

Section 11.6 found an equation for vmax of a rocket fired in deep space. What is vmax 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 ∆Py, the change of momentum in the vertical direction, in terms of dm and dvy. ∆Py 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 dvy. Then integrate to find a modified expression for vmax at the instant all the fuel has been burned.

a. What is vmax for a vertical launch from an airless planet ? Your answer will be in terms of mR, the empty rocket mass; mF0, the initial fuel mass; vex, 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?

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Step by Step Solution

Step 1. Given Information isMass of Rocket with fuel = 330,000 kgMass of fuel = 280,000 kgTime = 250 sEngine generates 4.1 MN of thrust 

We need to find out :

a) vmax 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

Step 2. Part a) Calculating vmax considering that gravity delivers an impulse

Now dividing equation1 by dt both sides,

Step 3. Part b) Rocket’s speed at the instant all the fuel has been burned if it is launched in deep space 

Mass of rocket with fuel, mo = 330,000 kg

Mass of fuel, mf = 280,000 kg

Mass of rocket, mr = 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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