A satellite moves in a circular orbit around
the Earth at a speed of 5.4 km/s.
Determine the satellite’s altitude above
the surface of the Earth. Assume the
Earth is a homogeneous sphere of radius
6370 km and mass 5.98 × 10
24
kg. The
value of the universal gravitational constant
is 6.67259 × 10
−11
N · m2
/kg
2
.
Answer in units of km
Why did the satellite go to therapy? Because it had some orbit issues! Now, let's calculate its altitude, shall we?
To determine the satellite's altitude above the surface of the Earth, we can use the centrifugal force and the force of gravity.
The centrifugal force acting on the satellite is equal to the gravitational force pulling it towards the Earth. Mathematically, these forces can be expressed as:
m * v^2 / r = G * M * m / (r + h)^2
Where:
m = mass of the satellite (which we can cancel out)
v = speed of the satellite (5.4 km/s)
r = radius of the Earth (6370 km)
G = universal gravitational constant (6.67259 × 10^-11 N · m^2 / kg^2)
M = mass of the Earth (5.98 × 10^24 kg)
h = altitude above the Earth's surface (what we're trying to find)
Now, let's plug in the values and solve for h:
(5.4 km/s)^2 / 6370 km = (6.67259 × 10^-11 N · m^2 / kg^2) * (5.98 × 10^24 kg) / (6370 km + h)^2
Let's do some calculations...
(29.16 km^2/s^2) / 6370 km = (3.996662542 × 10^14 N · m^2 / kg) / (6370 km + h)^2
Simplifying further...
29.16 / 6370 = 3.996662542 × 10^14 / (6370 + h)^2
Now, let's cross-multiply...
29.16 * (6370 + h)^2 = 3.996662542 × 10^14 * 6370
Expanding and simplifying...
(6370 + h)^2 = (3.996662542 × 10^14 * 6370) / 29.16
Taking the square root...
6370 + h = sqrt[(3.996662542 × 10^14 * 6370) / 29.16]
Solving for h...
h = sqrt[(3.996662542 × 10^14 * 6370) / 29.16] - 6370
After performing the calculations, the satellite's altitude above the surface of the Earth is approximately 1994 km. Ta-da! I hope that answer orbits around your expectations!
To determine the satellite's altitude above the surface of the Earth, we can use the equation for centripetal force:
F = (mv^2) / r
Where:
F is the gravitational force between the satellite and the Earth,
m is the mass of the satellite,
v is the orbital velocity of the satellite,
r is the distance from the center of the Earth to the satellite's center.
We can calculate the mass of the satellite using the equation:
m = F / g
Where:
g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
First, let's convert the orbital velocity from km/s to m/s:
v = 5.4 km/s * (1000 m/km)
v = 5400 m/s
Next, let's calculate the gravitational force using Newton's law of universal gravitation:
F = (G * m * M) / r^2
Where:
G is the gravitational constant,
M is the mass of the Earth, and
r is the sum of the Earth's radius and the satellite's altitude.
Now, we solve for the satellite's altitude:
r = (F * r^2) / (G * m * M)
r = (F * (r + 6370 km)^2) / (G * m * M)
Rearranging the equation, we have:
(F * r^2) / (G * m * M) - r = (F * (r + 6370 km)^2) / (G * m * M)
(1/GM) * (F * r^2 - G * m * M * r) = (F * r^2 + 2 * F * r * 6370 km + F * (6370 km)^2)
(1/GM) * (- G * m * M * r) = 2 * F * r * 6370 km + F * (6370 km)^2
Now, let's solve this equation step-by-step:
To determine the satellite's altitude above the surface of the Earth, we can use the concept of centripetal force and the gravitational force that keeps the satellite in orbit.
The centripetal force required to keep an object in circular motion can be calculated using the following formula:
Fc = (m * v^2) / r
where Fc is the centripetal force, m is the mass of the satellite, v is its velocity, and r is the radius of the orbit.
In this case, we know the speed of the satellite, which is given as 5.4 km/s. Since we don't know the mass of the satellite, we can use the value of gravitational force to calculate it.
The gravitational force between the satellite and the Earth can be calculated using the formula:
Fg = (G * m * M) / r^2
where Fg is the gravitational force, G is the universal gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth to the satellite, which is the sum of the Earth's radius (6370 km) and the satellite's altitude.
Setting the centripetal force equal to the gravitational force, we have:
(m * v^2) / r = (G * m * M) / r^2
Simplifying the equation, we find:
v^2 = (G * M) / r
Now we can substitute the given values into the equation:
(5.4 km/s)^2 = (6.67259 × 10^(-11) N·m^2/kg^2 * (5.98 × 10^24 kg)) / (6370 km + altitude)^2
Solving for altitude, we can rearrange the equation:
altitude = ((6.67259 × 10^(-11) N·m^2/kg^2 * (5.98 × 10^24 kg)) / (5.4 km/s)^2)^(1/2) - 6370 km
Now we can calculate the altitude by plugging in the values:
altitude = ((6.67259 × 10^(-11) N·m^2/kg^2 * (5.98 × 10^24 kg)) / (5.4 km/s)^2)^(1/2) - 6370 km
Calculating this, we find that the satellite's altitude above the surface of the Earth is approximately 426 km.
Therefore, the answer is 426 km.