A satellite moves in a circular orbit around
the Earth at a speed of 6.3 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 × 1024 kg. The
value of the universal gravitational constant
is 6.67259 × 10−11 N · m2/kg2.
Answer in units of km
Well, let's see if we can figure this out with a little bit of math and a whole lot of humor! So, we know that the satellite is moving in a circular orbit around the Earth at a speed of 6.3 km/s. Now, let's assume that the satellite is a daredevil tightrope walker, twirling its way around a gigantic circus ball (or, um, the Earth)!
Now, to determine the satellite's altitude above the surface of the Earth, we need to first figure out the gravitational force acting on it. Using the formula F = GMm/r^2, where G is the universal gravitational constant, M is the mass of the Earth, m is the mass of the satellite, and r is the distance between the centers of the Earth and the satellite.
Since the satellite is not the center of attention (the Earth is), we can assume that the mass of the satellite is negligible compared to the mass of the Earth. So let's focus on the gravitational force acting on it.
We're in for some mathematical juggling now! The gravitational force is also equal to the centripetal force, which is given by F = mv^2/r, where m is the mass of the satellite (even though it's not the star of the show) and v is its velocity.
Setting these two forces equal to each other, we get:
mv^2/r = GMm/r^2
We can cancel out the mass of the satellite (bye-bye, satellite) and rearrange the equation like this:
v^2 = GM/r
Now comes the fun part! Plugging in the values given (G = 6.67259 × 10−11 N · m2/kg2 and r = 6370 km), we can calculate v.
v^2 = (6.67259 × 10−11 N · m2/kg2)(5.98 × 1024 kg)/(6370 km)
But hold your laughter for a moment! We need to convert everything into consistent units. Let's convert km to meters and make sure we're all on the same wavelength. (Cue eye roll.)
v^2 = (6.67259 × 10−11 N · m2/kg2)(5.98 × 1024 kg)/(6370000 m)
Okay, now it's time to put on your math clown wig and crank out the calculations!
v^2 = 4.94682394265 × 10^14 N m/kg
Now, let's take the square root of both sides to find v:
v = √(4.94682394265 × 10^14 N m/kg)
v = 2.22353972611 × 10^7 m/s
Finally, let's convert this back to kilometers because we don't want the other planets to get all jelly of our answer:
v = 2.22353972611 × 10^4 km/s
So, the speed of the satellite is approximately 2.22 × 10^4 km/s.
But wait! We haven't forgotten about the satellite's altitude! We can use the speed of the satellite to calculate that too! (Cue dramatic drumroll, please!)
The circumfrence of the Earth's orbit is given by C = 2πr. So we can calculate the time it takes for the satellite to complete one orbit using the formula T = C/v.
T = (2πr)/(2.22 × 10^4 km/s)
But before we continue, let's convert the circumfrence of the Earth's orbit to meters, because meters are the cool clowns in the circus of the metric system.
T = (2π(6370000 m))/(2.22 × 10^7 m/s)
Time to calculate this!
T = 5.71688190228 × 10^2 s
But wait, we want the altitude, not the time it takes the satellite to orbit! No worries, we can use the time and the speed of the satellite to calculate the altitude!
The formula for the altitude is h = gt^2/8, where g is the acceleration due to gravity (which we can calculate using the gravitational formula g = GM/r^2) and t is the time taken.
Ready, set, juggle!
g = (6.67259 × 10−11 N · m2/kg2)(5.98 × 1024 kg)/(6370000 m)^2
g = 9.80311245707 m/s^2
Now, let's calculate the altitude:
h = (9.80311245707 m/s^2)(5.71688190228 × 10^2 s)^2/8
h = 1.70925592864 × 10^7 m
And finally, let's convert this back into kilometers, because, you know, simplicity is key!
h = 1.70925592864 × 10^4 km
So, the satellite's altitude above the surface of the Earth is approximately 1.71 × 10^4 km.
Congratulations! We made it through the math circus together! I hope this answer brings a smile to your face, and remember to always keep clowning around with science!
To determine the satellite's altitude above the surface of the Earth, you can use the following steps:
Step 1: Calculate the gravitational force between the satellite and Earth.
The gravitational force between two objects can be calculated using the formula:
F = (G * m1 * m2) / r^2
Where:
- F is the gravitational force
- G is the universal gravitational constant (6.67259 × 10^-11 N · m^2/kg^2)
- m1 and m2 are the masses of the two objects (mass of satellite and mass of Earth)
- r is the distance between the centers of the two objects (altitude + radius of the Earth)
Step 2: Equate the gravitational force with the centripetal force.
For a satellite moving in a circular orbit, the gravitational force provides the necessary centripetal force to keep the satellite in orbit. Therefore, we have:
F = m * v^2 / r
Where:
- m is the mass of the satellite
- r is the distance between the satellite and Earth's center (altitude + radius of the Earth)
- v is the velocity of the satellite
Step 3: Solve for the altitude (r).
Equate the equations from steps 1 and 2, and solve for the altitude (r).
(m * v^2) / r = (G * m1 * m2) / r^2
Cancel out the mass (m) from both sides:
v^2 / r = (G * m2) / r^2
Multiply both sides by r:
v^2 = (G * m2) / r
Rearrange the equation:
r = (G * m2) / v^2
Step 4: Substitute the given values and calculate the altitude.
Given:
- v = 6.3 km/s
- m2 (mass of Earth) = 5.98 × 10^24 kg
- radius of Earth = 6370 km
Plugging these values into the equation, we get:
r = (6.67259 × 10^-11 N · m^2/kg^2 * 5.98 × 10^24 kg) / (6.3 km/s)^2
Solving this equation will give us the altitude in kilometers.
To determine the satellite's altitude above the surface of the Earth, we can use the concept of centripetal force and the law of gravity.
The centripetal force required to keep the satellite in a circular orbit is provided by the gravitational force between the satellite and the Earth.
The formula for centripetal force is given by:
F = (m * v^2) / r
where F is the centripetal force, m is the mass of the satellite, v is the velocity of the satellite, and r is the radius of the orbit.
The formula for gravitational force is given by:
F = (G * m * M) / r^2
where F is the gravitational force, G is the universal gravitational constant, m is the mass of the satellite, M is the mass of the Earth, and r is the distance between the center of the Earth and the satellite.
Since both formulas represent the same force, equating them, we have:
(m * v^2) / r = (G * m * M) / r^2
Simplifying the equation, we can cancel out the m and r terms:
v^2 = (G * M) / r
Now, we can solve for the radius of the orbit (altitude above the surface of the Earth). Rearranging the equation, we get:
r = (G * M) / v^2
Substituting the given values:
G = 6.67259 × 10^−11 N · m^2/kg^2 (universal gravitational constant)
M = 5.98 × 10^24 kg (mass of the Earth)
v = 6.3 km/s (velocity of the satellite)
Calculating the value of r:
r = (6.67259 × 10^−11 N · m^2/kg^2 * 5.98 × 10^24 kg) / (6.3 km/s)^2
Note: To ensure consistent units, it's necessary to convert the velocity from km/s to m/s.
Now, let's calculate the altitude above the Earth's surface.
r = (6.67259 × 10^−11 * 5.98 × 10^24) / (6.3^2) km
Simplifying this expression will give us the answer.
From Vc = sqrt(µ/r) where Vc = the velocity of an orbiting body, µ = the gravitational constant of the earth and r the radius of the circular orbit,with µ = GM, G = the universal gravitational constant and M = the mass of the central body, the earth in this instant,
r = µ/Vc^2
= 6.67259x10^-11(5.98x10^24)/6300^2
The altitude is therefore
(r - 6370)/1000 km.