Verified answer

you can solve this with newtonian mechanics (simple, high school or lower) or with celestial mechanics (which are derived from the same principles, but vastly different notation). I'm assuming this is high school so this is what you have:

sum of forces in radial direction = m1*a = Force of gravity.

since its circular motion, a = v^2/r

force of gravity = G*m1*me/r^2

solving this gives v = sqrt(G*me/r)

altitude = 284 gives your r = 284+6378 = 6662 km

G = 6.67*10^(-11)

me = mass of earth = 5.97E24 kg

which gives v = 7.731 km/s

A Titan IV rocket has put your spacecraft in circular orbit around Earth at an altitude of 284 km. What is your orbital velocity? Give your answer in m/s with 4 numbers after the decimal.

The force of Universal gravitational attraction attracts any 2 objects towards each other.

Fg = (G * M1 * M2) ÷ R^2

G = 6.67 * 10^-11

M1 = mass of large object

M2 = mass of small object

R = distance between the center of mass of the 2 objects.

Altitude is the distance above the surface of the earth.

So, you need to know the distance from the center of the earth to the surface = 6.38 * 10^6 m.

R = 6.38 * 10^6 m + 284,000 m = 6.664 * 10^6 m

M1 = mass of Earth = 5.98 * 10^24 kg

Since the rocket is moving in a circle, the force must equal the centripetal force. The rocket is the small object = M2

Fc = M2 * v^2/R

Set these 2 equations equal to each other and solve for velocity!

M2 * v^2/R = (G * M1 * M2) ÷ R^2

Multiply both sides by R/M2

v^2 = (G * M1) ÷ R

v = [(G * M1) ÷ R]^0.5

Velocity = [(6.67 * 10^-11 * 5.98 * 10^24) ÷ (6.664 * 10^6)^2]^0.5

Velocity = 8.982 m/s