A satellite is in a circular orbit about the earth (ME = 5.98 1024 kg). The period of the satellite is 4.20 104 s. What is the speed at which the satellite travels?
T=2•π•r/v=2•π•sqrt(r³/GM)
T²=4π²r³/GM,
r=cuberoot{GMT²/4π²}=....
v=sqrt(GM/r)=.....
To find the speed at which the satellite travels, we can use the formula for the speed of an object in circular motion.
The formula for the speed of an object in circular motion is:
v = 2πr / T
Where,
v is the speed of the satellite,
π is the mathematical constant pi (approximately 3.14159),
r is the radius of the circular orbit,
T is the period of the satellite.
In this case, we are given the period of the satellite as 4.20 * 10^4 s.
To find the radius of the circular orbit, we need to use the formula for the period of an object in circular motion:
T = 2π√(r³ / GM)
Where,
T is the period of the satellite,
π is the mathematical constant pi (approximately 3.14159),
r is the radius of the circular orbit,
G is the gravitational constant (approximately 6.67430 * 10^-11 N m^2 / kg^2),
M is the mass of the Earth (approximately 5.98 * 10^24 kg).
Rearranging the formula to solve for r, we get:
r = (T²GM / 4π²)^(1/3)
Plugging in the known values, we get:
r = ((4.20 * 10^4 s)² * (6.67430 * 10^-11 N m^2 / kg^2) * (5.98 * 10^24 kg) / (4 * (3.14159)^2))^(1/3)
Simplifying the expression, we find:
r ≈ 6,678,696 meters
Now, plugging the values of r and T into the formula for velocity, we get:
v = (2π * 6,678,696 m) / (4.20 * 10^4 s)
Calculating the expression, we find:
v ≈ 29,878 m/s
Therefore, the speed at which the satellite travels is approximately 29,878 m/s.