A rocket with a mass of 2.5 × 105 kg is approaching straight into a relatively small black hole with the mass of the sun, MB = 1.99 × 1030kg. The pilot spots the black hole 1.496×1012m away and asks his physicist on board to do the following calculations:
a) What is the gravitational potential of at the location of the rocket?
b) What speed must the rocket reach to simply orbit the black hole and not fall in?
c) Determine the minimum distance from the black hole in which light cannot even escape,
the Schwarzchild radius. Consider the speed of light to be 3.00×108m/s
To calculate the answers to the given questions, you can utilize the equations related to gravitational potential, escape velocity, and the Schwarzschild radius. Let's go through each question step by step:
a) Gravitational Potential:
The formula for gravitational potential is given by:
Gravitational potential (V) = - (G * M) / r
Where:
G is the gravitational constant (6.674 × 10^-11 N m^2 / kg^2)
M is the mass of the black hole (1.99 × 10^30 kg)
r is the distance between the rocket and the black hole (1.496 × 10^12 m)
By substituting the given values in the formula, you can calculate the gravitational potential at the location of the rocket.
b) Orbit Velocity:
The speed required for an object to orbit around another object without falling in is called the orbit velocity (v). It can be calculated using the formula:
Orbit velocity (v) = √((G * M) / r)
By substituting the given values of G, M, and r into the formula, you can determine the speed the rocket must reach to simply orbit the black hole.
c) Schwarzschild Radius:
The Schwarzschild radius (Rs) is the radius at which the gravitational pull of an object becomes strong enough that light cannot escape from it. It can be calculated using the formula:
Schwarzschild radius (Rs) = (2 * G * M) / c^2
Where:
c is the speed of light (3.00 × 10^8 m/s)
By substituting the given values of G and M into the formula, you can calculate the minimum distance from the black hole in which light cannot escape.
Please note that it is always important to double-check the equations and units to ensure accuracy in your calculations.