A satellite moves on a circular earth orbit that has a radius of 6.68E+6 m. A model airplane is flying on a 16.3 m guideline in a horizontal circle. The guideline is nearly parallel to the ground. Find the speed of the plane such that the plane and the satellite have the same centripetal acceleration.
To find the speed of the plane such that it has the same centripetal acceleration as the satellite, we can use the formula for centripetal acceleration:
a = v^2 / r
where a is the centripetal acceleration, v is the speed of the object, and r is the radius of the circular motion.
For the satellite, the radius of the circular orbit is given as 6.68E+6 m.
For the plane, since it is flying on a 16.3 m guideline in a horizontal circle and the guideline is nearly parallel to the ground, the radius of the circular motion is 16.3 m.
Now, we can set up the equation to find the speed of the plane:
a_plane = a_satellite
v_plane^2 / r_plane = v_satellite^2 / r_satellite
We want to find v_plane.
Substituting the given values:
v_plane^2 / 16.3 = v_satellite^2 / 6.68E+6
We can rearrange the equation to solve for v_plane:
v_plane = sqrt(v_satellite^2 * 16.3 / 6.68E+6)
To find v_satellite, we need to use the concept of acceleration due to gravity and the formula for orbital speed:
v_satellite = sqrt(G * M / r_satellite)
where G is the gravitational constant, M is the mass of the Earth, and r_satellite is the radius of the satellite's orbit.
Given values:
G = 6.67430E-11 m^3 kg^-1 s^-2 (gravitational constant)
M = 5.9722E+24 kg (mass of the Earth)
Substituting the given values:
v_satellite = sqrt(6.67430E-11 * 5.9722E+24 / 6.68E+6)
Now we can substitute the value of v_satellite into the equation for v_plane to get the final answer.