A neutron star has a mass of 1.11 × 1030 kg and a radius of 7.93 × 103 m. Suppose an object falls from rest near the surface. How fast would it be moving after it had fallen a distance of 0.012 m? (Assume that the gravitational force is constant over the distance of the fall, and that the star is not rotating.)

To find the final velocity of an object falling near the surface of a neutron star, we can use the law of conservation of energy.

Given:
Mass of neutron star (M) = 1.11 × 10^30 kg
Radius of neutron star (R) = 7.93 × 10^3 m
Distance fallen (h) = 0.012 m

1. We can calculate the gravitational potential energy (PE) of the object at the initial position using the formula:

PE_initial = mgh

Where:
m = mass of the object (unknown)
g = acceleration due to gravity (constant near the surface of the neutron star)
h = distance fallen

2. Next, we can calculate the gravitational potential energy at the final position, which will be converted to kinetic energy:

KE_final = PE_initial

3. The kinetic energy (KE) can be calculated using the formula:

KE = (1/2) mv^2

Where:
m = mass of the object (unknown)
v = final velocity of the object

4. Equating KE_final and KE, we can solve for v:

(1/2) mv^2 = PE_initial

Substituting the value of PE_initial from step 1:

(1/2) mv^2 = mgh

5. We can now solve for v:

v^2 = 2gh

Taking the square root of both sides:

v = sqrt(2gh)

6. Substituting the given values:

g = G * M / R^2

Where:
G = gravitational constant
M = mass of the star
R = radius of the star

Let's calculate the final velocity.