Imagine you are in a spaceship one million km from a black hole of mass 2×1030 kg. Initially you are flying directly away from it at a velocity of 12.8 km/s.
Using the n-body approximation, what is your velocity one second later? Type your answer in metres per second?
To calculate the new velocity one second later, we need to use Newton's law of gravitation and the n-body approximation.
Newton's law of gravitation states that the gravitational force between two objects is given by:
F = (G * m1 * m2) / r^2,
where F is the gravitational force, G is the gravitational constant (6.67430 × 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between them.
In the n-body approximation, we assume that the mass of the spaceship is negligible compared to the black hole's mass. Therefore, the spaceship is only affected by the gravitational force from the black hole.
Given:
- Mass of the black hole (m2) = 2 × 10^30 kg
- Initial distance from the black hole (r) = 1 million km = 1 × 10^9 m
- Initial velocity of the spaceship (v) = 12.8 km/s = 12.8 × 10^3 m/s
First, let's calculate the initial gravitational force acting on the spaceship using Newton's law of gravitation:
F_initial = (G * m1 * m2) / r^2,
Since we are only interested in the acceleration and velocity of the spaceship, we will consider the force acting on it. The force will be given by:
F_initial = m1 * a_initial = (G * m1 * m2) / r^2,
where a_initial is the initial acceleration of the spaceship.
Rearranging the equation, we can solve for the acceleration:
a_initial = (G * m2) / r^2,
Using the given values:
a_initial = (6.67430 × 10^-11 N m^2/kg^2 * 2 × 10^30 kg) / (1 × 10^9 m)^2,
a_initial ≈ 2.669720 × 10^(-11) m/s^2.
Since acceleration is the rate of change of velocity, we can calculate the change in velocity (Δv) in one second as:
Δv = a_initial * t,
where t is the time interval, which in this case is 1 second.
Substituting the values:
Δv = 2.669720 × 10^(-11) m/s^2 * 1 s,
Δv ≈ 2.669720 × 10^(-11) m/s.
Finally, to calculate the new velocity (v_new), we add the change in velocity to the initial velocity:
v_new = v + Δv,
v_new = 12.8 × 10^3 m/s + 2.669720 × 10^(-11) m/s,
v_new ≈ 12.8000000266972 × 10^3 m/s.
Therefore, the new velocity of the spaceship one second later, using the n-body approximation, is approximately 12.8000000266972 × 10^3 m/s.