While visiting Planet Physics, you toss a rock straight up at 13.6 m/s and catch it 3.50 s later. While you visit the surface, your cruise ship orbits at an altitude equal to the planet's radius every 340.0 minutes.
What is the mass (in kg) of the planet?
The time of flight of the tossed rock is T = 2 Vo/g = 3.50 s.
Therefore the local value of g is
2 Vo/T = 7.77 m/s^2
The value of g is related to the radius R and mass M of the planet by
g = G*M/R^2
From the period of the orbit, you know that the orbital velocity is
V = 2*pi*R/P
where P is the known period, 20,400 s.
Since m V^2/R = m g,
4*pi^2*R/P^2 = 7.77 m/s^2
That equation can be solved for the orbit radius, R.
Then use g = G*M/R^2 for M
G is the universal constant of gravity,
6.67*10^-11 N m^2/kg^2
To determine the mass of the planet, we can use the fact that the gravitational force acting on the rock when it is thrown upwards is equal to the gravitational force when it is caught.
The gravitational force acting on an object can be calculated using Newton's law of gravitation:
F = G * (m1 * m2) / r^2
where:
F is the gravitational force
G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2/kg^2)
m1 and m2 are the masses of the two objects
r is the distance between the centers of the objects
In our case, the two objects are the rock and the planet. When the rock is thrown upwards, the distance between them is equal to the radius of the planet, denoted as r.
The equation for the gravitational force acting on the rock when thrown upwards is:
F1 = G * (m_rock * m_planet) / r^2
where m_rock is the mass of the rock and m_planet is the mass of the planet.
When the rock is caught by you, the distance between the rock and the planet is twice the radius, denoted as 2r. The equation for the gravitational force acting on the rock when caught is:
F2 = G * (m_rock * m_planet) / (2r)^2
Since the gravitational force is the same in both cases, we have:
F1 = F2
Therefore, we can set up the following equation:
G * (m_rock * m_planet) / r^2 = G * (m_rock * m_planet) / (2r)^2
Now, we can simplify the equation. Canceling out G and m_rock from both sides:
m_planet / r^2 = m_planet / (2r)^2
We can further simplify the equation:
1 / r^2 = 1 / (2r)^2
Now, we can solve for the mass of the planet:
m_planet = (2r)^2 / r^2
Given the information, the altitude is equal to the planet's radius, so we can substitute r with the altitude.
m_planet = (2 * altitude)^2 / altitude^2
Substituting the given altitude, we can calculate the mass of the planet.