A satellite is in a circular orbit 810 km above the Earth’s surface; i.e., it moves on a circular path under the influence of nothing but the Earth’s gravity.
Find the speed of the satellite. The radius of the Earth is 6.37 × 106 m, and the accel- eration of gravity at the satellite’s altitude is 7.7401 m/s2 .
Answer in units of km/s.
004 (part 1 of 2) 10.0 points
To find the speed of the satellite, we can use the formula for the centripetal acceleration of an object in circular motion:
ac = v²/r
Where ac is the centripetal acceleration, v is the velocity or speed of the object, and r is the radius of the circular path.
Given:
Radius of the Earth (rE) = 6.37 × 10^6 m
Acceleration of gravity at the satellite's altitude (g) = 7.7401 m/s²
We can calculate the radius of the orbit (rO) by adding the altitude of the satellite to the radius of the Earth:
rO = rE + altitude
altitude = 810 km = 810 × 10^3 m
rO = 6.37 × 10^6 m + 810 × 10^3 m
= (6.37 + 0.81) × 10^6 m
= 7.18 × 10^6 m
Now, we can calculate the speed of the satellite by rearranging the formula for centripetal acceleration:
v = √(ac * r)
Given:
ac = g = 7.7401 m/s²
r = rO = 7.18 × 10^6 m
v = √(7.7401 m/s² * 7.18 × 10^6 m)
v = √(55.5923598 × 10^6 m²/s²)
v = 7.46 × 10^3 m/s
Finally, we convert the speed from meters per second to kilometers per second:
v = 7.46 × 10^3 m/s * (1 km / 1000 m)
v = 7.46 km/s
Therefore, the speed of the satellite is approximately 7.46 km/s.
To find the speed of the satellite, we can use the formula for circular motion:
v = √(g * r)
where v is the speed of the satellite, g is the acceleration due to gravity at the satellite's altitude, and r is the distance between the center of the Earth and the satellite.
Given:
Acceleration due to gravity at the satellite's altitude (g) = 7.7401 m/s^2
Distance between the center of the Earth and the satellite (r) = radius of the Earth + altitude of the satellite
= (6.37 * 10^6 m) + (810 km) = (6.37 * 10^6 m) + (810 * 10^3 m)
Now let's substitute these values into the formula:
v = √(7.7401 m/s^2 * [(6.37 * 10^6 m) + (810 * 10^3 m)])
Simplifying the expression within the square root:
v = √(7.7401 m/s^2 * [6.37 * 10^6 + 810 * 10^3] m)
v = √(7.7401 m/s^2 * 6.37 * 10^6 m + 7.7401 m/s^2 * 810 * 10^3 m)
v = √(49.1669737 * 10^6 m^2/s^2 + 6.2837081 * 10^6 m^2/s^2)
Adding the terms within the square root:
v = √(55.4506818 * 10^6 m^2/s^2)
Taking the square root:
v = 7.449 m/s
Lastly, we need to convert the speed from m/s to km/s:
v = 7.449 m/s * (1 km / 1000 m)
v ≈ 0.007 km/s
Therefore, the speed of the satellite is approximately 0.007 km/s.