College Physics w/Calculus
posted by Oriana .
(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
(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