Sputnik, the first artificial satellite to orbit the Earth, had a mass of 83.6 kg and travelled at 7574 m/s. The radius of the earth is 6371 km and its mass is 5.972 x 10^24 kg. (NOTE: You may find different numbers online since there was some uncertainty in the exact orbit. These numbers were chosen to keep this question consistent. Also, this question requires a subtraction, which can cause difficulty with the numerical tolerance. Keep about 5 figures as you go through the calculation. I have increased the tolerance to 10%).
How high was Sputnik above the ground in meters during its orbit?
To find the height of Sputnik above the ground during its orbit, we first need to calculate the gravitational force acting on it.
The formula for gravitational force is:
F = (G * m1 * m2) / r^2
Where:
F = gravitational force
G = gravitational constant (6.67430 x 10^-11 m^3/kg/s^2)
m1 = mass of object 1 (Sputnik in this case)
m2 = mass of object 2 (Earth in this case)
r = distance between the centers of the two objects (radius of the Earth plus the height of Sputnik)
Let's begin by converting the radius of the Earth from kilometers to meters:
Radius of the Earth in meters = 6371 km * 1000 m/km = 6,371,000 m
Now let's calculate the gravitational force:
F = (G * m1 * m2) / r^2
F = (6.67430 x 10^-11 m^3/kg/s^2 * 83.6 kg * (5.972 x 10^24 kg)) / (6,371,000 m + height)^2
To find the height at which Sputnik orbits, we need to set the gravitational force equal to the centripetal force:
F = m * v^2 / r
Where:
F = gravitational force
m = mass of Sputnik (83.6 kg)
v = velocity of Sputnik (7574 m/s)
r = distance between the center of the Earth and Sputnik (radius of the Earth plus the height of Sputnik)
Setting the two equations equal to each other, we have:
(G * m1 * m2) / (6,371,000 m + height)^2 = m * v^2 / (6,371,000 m + height)
Simplifying the equation:
(G * m2) / (6,371,000 m + height) = v^2
Solving for the height, we have:
(6,371,000 m + height) = (G * m2 / v^2)
height = (G * m2 / v^2) - 6,371,000 m
Substituting the given values into the equation:
height = (6.67430 x 10^-11 m^3/kg/s^2 * (5.972 x 10^24 kg)) / (7574 m/s)^2 - 6,371,000 m
Calculating the equation:
height = 4,221,080 meters
Therefore, Sputnik was approximately 4,221,080 meters above the ground during its orbit.