Calculus Applications in Physics 2 Homework 7 Due Friday, March 2, 2018 (Counts

Question

Calculus Applications in Physics 2 Homework 7 Due Friday, March 2, 2018 (Counts the equation: as two problems.) The motion of a rocket in the absence of other forces is governed by dv dt dM dt where M is the mass of the rocket (continually changing), v is its velocity,t is time, and u is the speed at which propellant is spewed out the back (relative to the rocket). If we include the gravity of a planet or other celestial object with mass Mp, we get where G is the gravitational constant and r is the distance the rocket is from the center of the planet Suppose a small spacecraft (perhaps surveying for minerals) blasts off from the surface of a spherical asteroid with radius-= 1.2×10° m and mass M,-2.5 x 1020 kg. The rocket has initial mass M= 79,000 kg, and ejects mass at a constant rate R = 9.5 kg/s, so that M = Mo-R·The propellant is ejected at a speed of 4300 m/s. If the rocket engines burn for a total of 200 s, calculate the speed and height of the rocket when the engines shut off. Show that this speed is insufficient to escape the gravitational pull of the asteroid. (Calculating escape velocity from a planet or other spherical celestial object is a conservation of mechanical energy problem.) How long do the engines have to burn in order for the ship to escape the asteroid? Your Mathematica file should include an explanation of your methods, and you should make it easy to follow

Explanation / Answer

Start with the given equation, Mdv/dt = -udM/dt - GMpM/r²

Give M = M0 - Rt -> dM/dt = -R

Therefore equation becomes

(M0 - Rt)dv/dt = uR - GMp(M0 - Rt)/r²

dv/dt = uR/(M0 - Rt) - GMp/r²

Integrate on both sides with v limits to be 0 to v, and t 0 to 200

v - 0 = uR*(-1/R)* (ln((M0 - Rt) - ln(M0)) -  GMp/r²

v = u * (ln((M0 - Rt) - ln(M0)) -  GMp/r² -------------------(1)

Plugin the numbers, you get

v = 4300*(11.2529 - 11.2772) - 2.32

v = 102.17m/s

To find height, h integrate once more

h = v*t

h = 102.17*200

h = 8804.7 m

Escape velocity, Ve = square root of (2*GMp/r)

plugin the numbers, you get

Ve = 166.7083 m/s

Since v < Ve, Rocket cannot escape the planet

To find how long the engines should run, set v = Ve in (1)

plugin the numbers and solve for t using mathematica.

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