A rocket of mass,m is located at an alttitude h meters above the surface of a planet of a mass,M and radius, R.
*write an expression for the gravitational pull on it.
*Deduce an expression for the gravitational potential and escape velocity at that location.
F = G m M/r^2 = G m M/(R+h)^2
potential = - G m M /(R+h)
(1/2) m v^2 = G m M/(R+h)
so
v^2 = 2 G M/(R+h)
To calculate the gravitational pull on the rocket, we need to consider the gravitational force between the rocket and the planet.
According to Newton's law of gravitation, the gravitational force between two objects is given by:
F = G * (m * M) / r^2
Where:
F is the gravitational force
G is the gravitational constant (approximately 6.67430 × 10^-11 N(m/kg)^2)
m is the mass of the rocket
M is the mass of the planet
r is the distance between the center of the rocket and the center of the planet
However, in this case, we need to consider that the rocket is at an altitude h above the planet's surface. So the distance between the rocket and the planet's center, r, will be the sum of the radius of the planet R and the altitude h:
r = R + h
Therefore, the expression for the gravitational pull on the rocket is:
F = G * (m * M) / (R + h)^2
To deduce an expression for the gravitational potential at that location, we need to consider the work done in bringing the rocket from infinity to that altitude h above the surface of the planet. The gravitational potential at a point is the work required to bring a unit mass from infinity to that point.
The expression for gravitational potential, V, is given by:
V = -G * (M / r)
Where:
V is the gravitational potential
G is the gravitational constant
M is the mass of the planet
r is the distance between the center of the rocket and the center of the planet (R + h)
Substituting r = R + h into the expression, the gravitational potential at that location becomes:
V = -G * (M / (R + h))
Finally, to calculate the escape velocity at that location, we need to equate the kinetic energy of the rocket to its potential energy at that point.
The expression for escape velocity, Ve, is given by:
Ve = sqrt((2 * G * M) / r)
Substituting r = R + h into the expression, the escape velocity at that location becomes:
Ve = sqrt((2 * G * M) / (R + h))