The space shuttle releases a satellite into a circular orbit 700 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the release occurs?
First you must find the Force of Gravity usin5g Fg=Gm1/r^2
G=6.67E-11
m1= mass of the earth: 5.9736E24
r= (radius of earth plus the radius of sattelite released)
radius of earth = 6,371,000m + 700,000m = 7,071,000
Fg=(6.67E-11)(5.9736E24)/(7071000^2)
Fg= 7.97
Next you must find the velocity necessary for centripetal acceleration to equal the force of gravity that you just found.
7.97=v^2/r
7.97=v^2/7,071,000
56355870=v^2
take the square root
v=7507 m/s
To determine the speed at which the space shuttle must be moving when releasing the satellite into a circular orbit, we can use the concept of centripetal force. The centripetal force required to keep an object in circular motion is provided by the gravitational force between Earth and the satellite.
Step 1: Find the gravitational force acting on the satellite.
The gravitational force formula is given by:
F = G * (m1 * m2) / r^2
where F is the force, G is the gravitational constant (6.674 × 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 this case, m1 is the mass of the satellite, m2 is the mass of the Earth, and r is the distance from the satellite to the center of the Earth.
Step 2: Equate the gravitational force with the centripetal force.
The centripetal force required to keep an object in circular motion is given by:
F = (m * v^2) / r
where F is the force, m is the mass of the satellite, v is the velocity, and r is the radius of the circular orbit.
Since we have already calculated the gravitational force, we can equate these two forces:
G * (m1 * m2) / r^2 = (m * v^2) / r
Step 3: Solve for v.
Rearranging the equation, we can solve for v:
v = sqrt(G * (m2/r))
Step 4: Calculate the velocity.
Plug in the values into the equation:
v = sqrt(6.674 × 10^-11 N m^2/kg^2 * (5.972 × 10^24 kg) / (700,000 m + 6,371,000 m))
By evaluating this equation, you will find that v is approximately 7,900 m/s (meters per second), which is the speed that the space shuttle must be moving (relative to Earth) when releasing the satellite into a circular orbit 700 km above the Earth.