At what speed must a satellite of a mass M be launched horizontally at the surface of the earth if it's orbit is to be a circle just grazing the highest mountains?( assume no air resistance). How much time would elapse between successive passes of this imaginary satillite?

Having been a member of the Southwestern Rocket Society in its infant days, we learned that launching rockets horizontally was dangerous.

At the altidude of the highest mountains, say 10^4 meters, then

the gravitational field strength g, is given by
9.8Nt/kg * (re/(re+10^4))^2
look up the radius of the earth re, in meters, and you have that part.
Then that gravational field strength mus be equal to centripetal acceleration, which is

acceleartion= V^2/(re+10^4) where
v is 2PI(re+10^4)/Period

set them equal, solve for Period.

To determine the speed at which a satellite must be launched, we need to consider the relationship between the gravitational force and the centripetal force.

Step 1: Find the altitude of the highest mountains.
The average altitude of the highest mountains on Earth is approximately 8,848 meters (29,029 feet). We can consider this as the altitude for our calculations.

Step 2: Find the radius of the orbit.
To calculate the radius of the orbit, we add the radius of the Earth (6,371 kilometers or 6,371,000 meters) to the altitude of the highest mountains.
Radius of orbit = Radius of Earth + Altitude of highest mountains
= 6,371,000 meters + 8,848 meters
= 6,379,848 meters.

Step 3: Find the acceleration due to gravity.
The acceleration due to gravity can be calculated using the formula:
acceleration due to gravity (g) = G *(Mass of Earth) / (Radius of Earth)^2,
where G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2).

Step 4: Find the speed of the satellite.
The centripetal force required to keep the satellite in orbit is provided by the gravitational force. Equating these forces, we have:
(G * Mass of Earth * Mass of Satellite) / (Radius of orbit)^2 = (Mass of Satellite * (Velocity of satellite)^2) / (Radius of orbit).

Simplifying the equation gives:
Velocity of the satellite = sqrt(G * Mass of Earth / Radius of orbit).

Step 5: Calculate the time between successive passes.
The time between successive passes can be calculated using the formula:
Time = (2π * Radius of orbit) / Velocity of satellite.

Now we can plug in the values into the formulas to find the solutions:

1. Calculate the acceleration due to gravity:
G = 6.67430 × 10^-11 m^3 kg^-1 s^-2 (gravitational constant)
Mass of Earth = 5.972 × 10^24 kg (mass of Earth)
g = G * Mass of Earth / (Radius of Earth)^2.
g = (6.67430 × 10^-11 m^3 kg^-1 s^-2 * 5.972 × 10^24 kg) / (6,371,000 meters)^2.

2. Calculate the radius of the orbit:
Radius of orbit = Radius of Earth + Altitude of highest mountains
= 6,371,000 meters + 8,848 meters
= 6,379,848 meters.

3. Calculate the speed of the satellite:
Velocity of the satellite = sqrt(G * Mass of Earth / Radius of orbit).

4. Calculate the time between successive passes:
Time = (2π * Radius of orbit) / Velocity of satellite.

By following these steps, you should be able to calculate the required speed and the time between successive passes for the satellite.

To determine the speed at which the satellite must be launched and the time between successive passes, we can use the concept of gravitational force and centripetal force.

1. Speed of the Satellite:
For a satellite in a circular orbit, the gravitational force between the satellite and the Earth provides the necessary centripetal force to keep it in orbit.
The centripetal force is given by:

F_c = (M * V^2) / R

Where,
F_c is the centripetal force,
M is the mass of the satellite,
V is the velocity of the satellite,
R is the radius of the orbit.

The gravitational force is given by:

F_g = (G * M * m) / R^2

Where,
F_g is the gravitational force,
G is the gravitational constant,
M is the Mass of the Earth,
m is the mass of the satellite,
R is the radius of the Earth.

In order for the satellite to just graze the highest mountains, the radius of its orbit (R) would be the sum of the radius of the Earth (R_E) and the height of the highest mountain (h). So:

R = R_E + h

Where,
R is the radius of the orbit.
R_E is the radius of the Earth.
h is the height of the highest mountain.

Setting the centripetal force equal to the gravitational force, we have:

(M * V^2) / (R_E + h) = (G * M * m) / R_E^2

Simplifying the equation, we can solve for V:

V = sqrt((G * M) / (R_E + h))

2. Time between Successive Passes:
The time period (T) of an orbiting satellite is the time it takes to complete one revolution around the Earth. It can be calculated using the formula:

T = (2 * π * R) / V

Where,
T is the time period,
R is the radius of the orbit,
V is the velocity of the satellite.

Substituting the value of R (R = R_E + h) and V into the equation, we can solve for T:

T = (2 * π * (R_E + h)) / sqrt((G * M) / (R_E + h))

This will give us the time between successive passes of the satellite.

Note: The calculations assume a circular orbit and no air resistance.

Now, to obtain the specific numerical values for the Earth's radius (R_E), the height of the highest mountain (h), the Earth's mass (M), and the gravitational constant (G), you will need to refer to reliable sources or use approximations based on the context of the problem.