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%).

Question

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?

GPT 3.5 · Answer

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.