# It is easy to integrate v f v i z t f t i dv dt dt u

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It is easy to integrate: v f - v i = Z t f t i dv dt dt = - u Z t f t i 1 m dm dt dt = - u Z m f m i 1 m dm = u ln m i m f For example, with u=100 m/s, a 100 kg rocket ejecting 10 kg gains in velocity Δ v = 100 ln 100 90 m/s = 10 . 5 m/s = 40 km/hr The thrust (force) on the rocket is u dm dt

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Phys 2A - Mechanics You are the ground-control commander of a 2000 kg scientific rocket that is approaching Mars at a speed of 2.5 × 10 4 km/hr. It needs to quickly slow to 1.13 × 10 4 km/hr to begin a controlled descent to the surface. If the rocket enters the Martian atmosphere too fast it will burn up, and if it enters too slowly, it will use up its maneuvering fuel before reaching the surface and will crash. The rocket has a new braking system: Several 5.0 kg “bullets” on the front of the rocket can be fired straight ahead. Each has a high-explosive charge that fires it at a speed of 139,000 m/s relative to the rocket. You need to send the rocket an instruction to tell it how many bullets to fire. Success will bring you fame and glory, but failure of this \$500,000,000 mission will ruin your career. How many bullets will you tell the rocket to fire? Take that the mass of the rocket is constant. Comment on the assignment: The last instruction “Take that the mass of the rocket is constant” is WRONG. To solve the problem you must fire the bullets sequentially and account for the decrease of the mass. You can use a spreadsheet (or compute many times). The rocket motion equation is not a good approximation.
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