(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

dingus

To find the answers to the second and third questions, we can use the concept of gravitational potential energy and apply the conservation of energy principle. Let's go step by step:

(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?

To find the distance from the surface the particle will go, we need to determine its maximum height or distance from the surface before it starts falling back towards the asteroid. We can do this by equating the initial kinetic energy of the particle to its final gravitational potential energy.

1. Calculate the initial kinetic energy (KE_initial):
KE_initial = (1/2) * mass * (speed)^2

2. Calculate the final gravitational potential energy (PE_final):
PE_final = mass * gravitational acceleration * distance

Since the kinetic energy is completely converted to potential energy at the maximum height or distance, we can equate them and solve for distance:

KE_initial = PE_final

Plug in the values, and rearrange the equation to solve for distance (d).

(c) With what speed will an object hit the asteroid if it is dropped from 1000 km above the surface?

To find the speed at which the object hits the asteroid, we need to determine the final speed just before impact. Again, we can use the concept of conservation of energy:

1. Calculate the initial gravitational potential energy (PE_initial):
PE_initial = mass * gravitational acceleration * initial height

2. Calculate the final kinetic energy (KE_final) just before impact:
KE_final = (1/2) * mass * (speed)^2

Since the potential energy is completely converted to kinetic energy just before impact, we can equate them and solve for speed:

PE_initial = KE_final

Plug in the values, and rearrange the equation to solve for speed (v).

By following these steps, you should be able to find the answers to the second and third questions.

chad