The time in hours it takes a satellite to complete an orbit around the earth varies directly as the radius of the orbit (from the center of the earth) and inversely as the orbital velocity. If a satellite completes an orbit 860 miles above the earth in 12 hours at a velocity of 34,000 mph, how long would it take a satellite to complete an orbit if it is at 1700 miles above the earth at a velocity of 27,000 mph? (Use 3960 miles as the radius of the earth.)
T1= r1/r2 * V2/V1
T1= (1700+3960)/(860+3960) * 34000/27000
The actual velocity of a satellite orbiting at 860 miles altitude is 24,113 fps.
The period derives from
T = 2(Pi)sqrt(r^3/µ) = 113.3 minutes.
The actual velocity of a satellite orbiting at 1700 miles altitude is 21,705 fps.
The period derives from
T = 2(Pi)sqrt(r^3/µ) = 144.18 minutes.
Alternatively,
T2/T1 = sqrt[(r2)^3/(r1)^3]
or T2/T1 =sqrt[5660^3/4820)^3]=1.272489
T2 = 113.3(1.272489) = 144.18
To solve this problem, we need to determine the relationship and set up a proportion based on the given information.
Step 1: Establish the direct and inverse relationships:
Let's denote:
r1 = radius of the first orbit (860 miles above the earth)
v1 = velocity of the first orbit (34,000 mph)
t1 = time it takes to complete an orbit in the first orbit (12 hours)
Let's also denote:
r2 = radius of the second orbit (1700 miles above the earth)
v2 = velocity of the second orbit (27,000 mph)
t2 = time it takes to complete an orbit in the second orbit (to be determined)
The problem states that the time it takes to complete an orbit varies directly with the radius and inversely with the velocity. So, we can write the following equations:
t1 ∝ r1 (direct relationship)
t1 ∝ 1/v1 (inverse relationship)
Similarly:
t2 ∝ r2 (direct relationship)
t2 ∝ 1/v2 (inverse relationship)
Step 2: Set up the proportion:
Based on the given information and the relationships established above, we can set up the following proportion:
t1/t2 = (r1/r2) * (v2/v1)
Step 3: Substitute values and solve for t2:
Let's substitute the known values into the proportion:
12/t2 = (860/1700) * (27,000/34,000)
Simplifying the proportion:
12/t2 = (43/85) * (27/34)
12/t2 = 1161/2,290
To solve for t2, we'll cross multiply:
1161 * t2 = 12 * 2,290
t2 = (12 * 2,290) / 1161
Calculating the result:
t2 ≈ 23.6
Therefore, it would take approximately 23.6 hours for the satellite to complete an orbit if it is at 1700 miles above the earth at a velocity of 27,000 mph.
To determine how long it would take a satellite to complete an orbit at a different altitude and velocity, we need to consider the relationships described in the problem.
Let's break down the information given:
1. The time it takes to complete an orbit varies directly with the radius of the orbit and inversely with the orbital velocity.
2. The satellite completes an orbit 860 miles above the earth in 12 hours at a velocity of 34,000 mph.
3. The radius of the Earth is 3960 miles.
We can set up a proportion to solve for the unknown time:
(time1) / (time2) = (radius2 / radius1) * (velocity1 / velocity2)
Let's fill in the known values:
(time1) / 12 = (1700 + 3960) / (860 + 3960) * 34000 / 27000
Simplifying this equation, we get:
(time1) / 12 = 5660 / 4820 * 34/27
Now, let's solve for (time1):
Cross-multiplying the equation, we get:
(time1) = 12 * (5660 / 4820) * (34 / 27)
Calculating this expression, we find:
(time1) ≈ 16.126 hours
Therefore, it would take approximately 16.126 hours for the satellite to complete an orbit at 1700 miles above the Earth with a velocity of 27,000 mph.