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College Physics w/Calculus

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(a) What is the escape speed on a spherical asteroid whose radius is 570 km and whose gravitational acceleration at the surface is 3.1 m/s2?
(b) How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 1000 m/s?
(c) With what speed will an object hit the asteroid if it is dropped from 1000 km above the surface?

I know how to do the first question (conservation of mechanical energy): however, I have no idea how to do the other two problems

  • College Physics w/Calculus -

    (a) Potential enegy increase = -M*Integral of g(r) dr (R to infinity) = -M*g(R)*R^2/r^2 dr
    = M*g(R)/R = (1/2) M Vo^2
    Vo = sqrt(2g(R)*R)
    = 1880 m/s
    is the escape velocity

    (b) PE change = M*[g(R)*R^2(1/R - 1/r) g(R)*R/r]
    = (1/2) M V^2

    2 g(R)* R^2 (1/R - 1/r) = V^2

    2g(R)*R - V^2 = 2g(R)*R^2/r
    = 3.53*10^6 - 10^6 m^2/s^2 = 2.53*10^6 m^2/s^2

    r = 2*(3.1 m/s^2)(570*10^3)^2 m^2/2.53*10^6 m^2/s^2
    = 7.96*10^5 m
    = 796 km. Subtract the asteroid radius (570 km) from that to get the altitude above the surface

    (c) The kinetic energy at R will equal the PE decrease in going from R + 1000 km to R altitude

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