A satellite is in a circular orbit about the earth (ME = 5.98 1024 kg). The period of the satellite is 1.30 104 s. What is the speed at which the satellite travels?
Equate the centripetal force to the gravity force and solve for the orbit radius, R. Then use
2 pi R/Period = speed
To find the speed at which the satellite travels in its circular orbit, we can use the equation:
v = 2πr/T
Where:
v is the speed of the satellite
r is the radius of the orbit
T is the period of the satellite
We know the period of the satellite is 1.30 x 10^4 s. We can calculate the radius of the orbit using the formula:
T = 2π√(r^3/GMe)
Where:
G is the gravitational constant (approximately 6.67 x 10^-11 m^3/kg/s^2)
Me is the mass of the Earth (approximately 5.98 x 10^24 kg)
Rearranging the formula, we get:
r = (T^2 * G * Me / (4π^2)) ^(1/3)
Substituting the given values, we can calculate the radius:
r = ( (1.30 x 10^4)^2 * 6.67 x 10^-11 * 5.98 x 10^24 / (4π^2)) ^(1/3)
After performing the calculation, we can substitute the value of r into the first equation to calculate the speed:
v = 2π * r / T
Substituting the calculated value of r and the given value of T, we can determine the speed at which the satellite travels.
To find the speed at which the satellite travels, we can use the formula for the circumference of a circle:
C = 2πr
Where C is the circumference and r is the radius of the orbit.
Since the satellite is in a circular orbit, the radius of the orbit is equal to the distance between the satellite and the center of the Earth. However, we don't have this information directly.
But we know that the period T (time taken for one complete revolution) and the radius r are related by Kepler's third law:
T^2 = (4π^2 / GM) * r^3
Here G is the gravitational constant and M is the mass of the Earth.
We can rearrange this equation to solve for r:
r = (T^2 * GM / (4π^2))^(1/3)
Now that we have the radius, we can find the circumference:
C = 2πr
Finally, to find the speed v of the satellite, we can divide the circumference C by the time period T:
v = C / T
Let's plug in the given values:
M = 5.98 × 10^24 kg
T = 1.30 × 10^4 s
G = gravitational constant = 6.67430 × 10^(-11) m^3 kg^(-1) s^(-2)
π = 3.14159
First, let's calculate the radius using the equation for r:
r = (T^2 * GM / (4π^2))^(1/3)
Now, calculate the value using the given values for G, M, and T:
r = ((1.30 × 10^4 s)^2 * (6.67430 × 10^(-11) m^3 kg^(-1) s^(-2)) * (5.98 × 10^24 kg) / (4(3.14159)^2))^(1/3)
After evaluating this expression, we get the radius r.
Next, let's calculate the circumference C:
C = 2πr
Now, plug in the value of r to calculate the circumference C.
Finally, let's calculate the speed v using the formula:
v = C / T
After plugging in the values of C and T, you will obtain the speed at which the satellite travels.