An earth satellite describes a circular orbit of radius R about the center of the earth. Calculate R if the orbit period is one day. Calculate initial vertical velocity to leave the satellite from the surface to a distance R from the center. Calculate launching velocity for a circular orbit, assuming earth's rotation is not considered.

Part (c): Velocity for circular orbit at radius R Step 1: Gravitational force provides the centripetal force required for circular motion. (G M m)/(R^2) = m v^2R Step 2: Cancel m from both sides. (G M)/(R^2) = v^2R Step 3: Multiply both sides by R. (G M)/(R) = v^2 Step 4: Take square root. v = sqrt((G M)/(R)) Final answer for (c): sqrt((G M)/(R)) Part (d): Initial vertical velocity from Earth's surface (radius R_e) to reach distance R from center Step 1: Use conservation of mechanical energy. Kinetic + potential energy at surface equals potential energy at maximum distance R (where velocity is zero). (1)/(2) m v_0^2 - (G M m)/(R_e) = -(G M m)/(R) Step 2: Divide both sides by m. (1)/(2) v_0^2 - (G M)/(R_e) = -(G M)/(R) Step 3: Add (G M)/(R_e) to both sides. (1)/(2) v_0^2 = (G M)/(R_e) - (G M)/(R) Step 4: Factor out G M. (1)/(2) v_0^2 = G M ( (1)/(R_e) - (1)/(R) ) Step 5: Multiply both sides by 2. v_0^2 = 2 G M ( (1)/(R_e) - (1)/(R) ) Step 6: Take square root. v_0 = sqrt(2 G M ( (1)/(R_e) - (1)/(R) )) Final answer for (d): sqrt(2 G M ( (1)/(R_e) - (1)/(R) ))