With what orbital speed will a satellite circle Jupiter if placed at a height of 8.50 * 10 ^ 6 is 1.90 * 10 ^ 27 kg and the radius of Jupiter is 7.14 * 10 ^ 7 * m .

Answer 1

Let's create a general problem solving strategy to help approach problems like these.

1. Set up an equation using our known variables, as well as the variable we're solving for.
2. Leaving the equation as variables, expand and cancel out like terms.
3. When the equation has been reduced, plug in the values for the variables. Including units isn't required, but will help you verify your work and keep things cleaner.

For this problem, we need an equation with velocity, orbital radius, and mass. My mind thinks of gravity and centripetal force. An object in orbit has equivalent gravitational and centripetal force

F_{g =} F_c

Now let's expand the two equations, with M_j being the mass of Jupiter, M_o being the mass of the satellite, r being the orbital radius, V being the linear orbital speed, and G being the gravitational constant. The orbital radius will be equal to the height of the object plus the radius of Jupiter.

(GM_JM_o) / r² = (M_oV²) / r

Cancel out like terms (M_o and an r) and simplify for V, our orbital speed

(G * M_{J *} M_o) / r² = (M_{o *} V²) / r

(G * M_J) / r = V²

V = sqrt(G * M_J / r)

Time to plug in the values.

G = 6.67*10^-11 N*m²/kg²

r = (8.50 * 10 ^ 6) + (7.14 * 10 ^ 7) = 7.99 * 10⁷ m

M_j = 1.90 * 10²⁷ kg

V = sqrt(6.67*10^-11 N*m²/kg² * 1.90 * 10²⁷ kg / 7.99 * 10⁷ m)

V = 39,826

To double check, we can simplify the units. Replace N with kg*m/s²

V = m/s = sqrt[ (N*m²/kg²)*(kg)/(m) ]

V = m/s = sqrt[ (kg*m/s²) * (m²/kg²) * (kg) / (m) ]

V = m/s = sqrt[ kg²*m³ / kg²*m*s² ]

V = m/s = sqrt[ m² / s² ]

V = m/s = m/s

V = 39,829 m/s

Our answer, in linear velocity, is 39,826 m/s. If we want the angular velocity, simply divide this number by the radius.

V = r * W

W = V / r

W = 39826 / 7.99 * 10⁷

W = 0.0005 radians/second

Answer 2

the orbital speed is

(GMm/r²)=mv²/r

=> v =√(GM/r)

=> v =√(GM/(R+h))

=> v= √((6.67*10^-11)x(1.9*10²⁷))/(0.86*10⁷+7.14*10⁷)

=> v = 3.98 * 10⁴ m/s