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? V = m/s ??
To find the speed at which the space shuttle must be moving when it releases the satellite into a circular orbit 700 km above the Earth, we can use the concept of centripetal force.
The centripetal force acting on the satellite is provided by the gravitational force between the Earth and the satellite. Therefore, we can equate these two forces:
F = (m * v^2)/r = G * (m * M)/r^2
Where:
F = Centripetal force (provided by the gravitational force)
m = Mass of the satellite
v = Velocity of the satellite
r = Radius of the satellite's orbit (700 km + radius of the Earth, which is approximately 6,370 km)
G = Gravitational constant (approximately 6.67430 × 10^-11 m^3⋅kg^−1⋅s^−2)
M = Mass of the Earth
We can cancel out the mass of the satellite from both sides of the equation, and rearrange the equation to solve for v:
v^2 = G * M/r
Now, plug in the values:
v^2 = (6.67430 × 10^-11 m^3⋅kg^−1⋅s^−2) * (5.972 × 10^24 kg) / (700 km + 6,370 km)
Simplified:
v^2 = (6.67430 × 10^-11 m^3⋅kg^−1⋅s^−2) * (5.972 × 10^24 kg) / (700,000 + 6,370,000) m
Calculating the right side:
v^2 ≈ (2.9645 × 10^5 m^2⋅s^−2)
Taking the square root of both sides to solve for v:
v ≈ 5443.26 m/s
Therefore, the space shuttle must be moving at approximately 5443.26 m/s (relative to Earth) when it releases the satellite into a circular orbit 700 km above the Earth.
To find the speed of the shuttle when the satellite is released, we can use the concept of orbital velocity.
Orbital velocity refers to the velocity at which an object must move in order to stay in a stable orbit around a larger body, such as the Earth. It is calculated using the formula:
V = sqrt(G * M / R)
where:
V is the orbital velocity,
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2),
M is the mass of the Earth (approximately 5.972 × 10^24 kg),
R is the distance between the center of the Earth and the satellite.
In this case, the satellite is released into a circular orbit 700 km above the Earth's surface. However, the formula for orbital velocity requires the distance to be measured from the center of the Earth. Therefore, we need to add the radius of the Earth (approximately 6,371 km) to the altitude of the satellite to get the value of R.
R = (700 km + 6,371 km) = 7,071 km = 7,071,000 m
Now, we can plug in the values into the formula to calculate the orbital velocity:
V = sqrt((6.67430 × 10^-11 m^3 kg^-1 s^-2) * (5.972 × 10^24 kg) / (7,071,000 m))
Simplifying this equation will give us the value of V, which represents the speed of the shuttle relative to the Earth when the satellite is released.