When a satellite is h kilometres above Earth,the time,t,in minutes,to complete one orbit is given by the formula t=Root of (6370+h)^3/6024
a) A telecommunications satellite is placed 30 km above Earth. How long does it take the satellite to make one orbit?
b) A satellite is placed in geosynchronous orbit about Earth. What must its altitude be?
*the satellite is travelling at more than 11,000 km/h
Please help me understand this. Thank you!!
a) To find the time it takes for the telecommunications satellite to make one orbit, which is represented by t, we need to substitute the given value for h into the formula and calculate the result.
Given:
h = 30 km
Formula:
t = √((6370 + h)^3/6024)
To calculate t, we substitute the value of h into the formula:
t = √((6370 + 30)^3/6024)
t = √(6400^3/6024)
To calculate this, we need a calculator or software that supports scientific notation. By calculating the value, we find that t ≈ 101.666 minutes.
Therefore, it takes approximately 101.666 minutes (or 1 hour and 41.67 minutes) for the telecommunications satellite to make one orbit.
b) To determine the altitude required for a satellite to be in geosynchronous orbit, we need to set the time, t, equal to 24 hours (1440 minutes). Then, we can solve for h.
Given:
t = 1440 minutes (time for one Earth day)
Formula:
t = √((6370 + h)^3/6024)
We substitute the value of t into the formula:
1440 = √((6370 + h)^3/6024)
Next, we can square both sides of the equation to eliminate the square root:
(1440^2) = (6370 + h)^3/6024
Then, multiply both sides by 6024 to get rid of the denominator:
6024 * (1440^2) = (6370 + h)^3
Now, we can find the cube root of both sides to isolate (6370 + h):
∛(6024 * (1440^2)) = 6370 + h
By calculating this value, we find that ∛(6024 * (1440^2)) ≈ 53818.47.
So, we can solve for h:
h ≈ 53818.47 - 6370
Thus, the altitude of a satellite in geosynchronous orbit is approximately 47448.47 km.
To determine if a satellite can be placed in geosynchronous orbit, we also need to consider the speed requirement. You mentioned that the satellite is traveling at more than 11,000 km/h. Geosynchronous orbit requires a speed that matches the rotation of the Earth, which is approximately 1670 km/h. Since the given speed is greater than the required speed, this satellite can be placed in geosynchronous orbit.