An earth satellite describes a circular orbit of radius R about the center of the earth. Calculate R if the…

Question

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.

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Part (c): Velocity for circular orbit at radius R Step 1: Gravitational force provides the centripetal force required for circular motion. (G M m)/(R²) = m (v²)/(R) Step 2: Cancel m from both sides. (G M)/(R²) = (v²)/(R) Step 3: Multiply both sides by R. (G M)/(R) = v² Step 4: Take square root. v = √((G M)/(R)) Final answer for (c): √((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 v₀² − (G M m)/(R_e) = −(G M m)/(R) Step 2: Divide both sides by m. 1/2 v₀² − (G M)/(R_e) = −(G M)/(R) Step 3: Add (G M)/(R_e) to both sides. 1/2 v₀² = (G M)/(R_e) − (G M)/(R) Step 4: Factor out G M. 1/2 v₀² = G M ( 1/R_e − 1/R ) Step 5: Multiply both sides by 2. v₀² = 2 G M ( 1/R_e − 1/R ) Step 6: Take square root. v₀ = √(2 G M ( 1/R_e − 1/R )) Final answer for (d): √(2 G M ( 1/R_e − 1/R ))

Answers

Step-by-step solution

Part c· 4 steps

Gravitational force provides the centripetal force required for circular motion.

Cancel $m$ from both sides.

Multiply both sides by $R$ .

Take square root.

Part d· 6 steps
$\sqrt{2 G M \left( \dfrac{1}{R_e} - \dfrac{1}{R} \right)}$

Use conservation of mechanical energy.

Divide both sides by $m$ .

Add $\frac{G M}{R_e}$ to both sides.

Factor out $G M$.

Multiply both sides by 2.

Take square root.

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Answers

Step-by-step solution

Part c· 4 steps

Gravitational force provides the centripetal force required for circular motion.

Cancel mmm from both sides.

Multiply both sides by RRR.

Take square root.

Part d· 6 steps2GM(1Re−1R)\sqrt{2 G M \left( \dfrac{1}{R_e} - \dfrac{1}{R} \right)}2GM(Re​1​−R1​)​

Use conservation of mechanical energy.

Divide both sides by mmm.

Add GMRe\frac{G M}{R_e}Re​GM​ to both sides.

Factor out $G M$.

Multiply both sides by 2.

Take square root.

Try one yourself

Still stuck on this one?

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Cancel $m$ from both sides.

Multiply both sides by $R$ .

Part d· 6 steps
$\sqrt{2 G M \left( \dfrac{1}{R_e} - \dfrac{1}{R} \right)}$

Divide both sides by $m$ .

Add $\frac{G M}{R_e}$ to both sides.