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What is the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R?
- a)2GmM/3R
- b)5GmM/6R
- c)GmM/2R
- d)GmM/3R
Correct answer is option 'B'. Can you explain this answer?
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What is the minimum energy required to launch a satellite of mass m fr...
From conservation of energy,
Total energy at the planet = Total energy at altitude
Total energy at the planet = Total energy at altitude
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What is the minimum energy required to launch a satellite of mass m fr...
Minimum Energy Required to Launch a Satellite in Circular Orbit
To launch a satellite of mass m in a circular orbit at an altitude of 2R from the surface of a planet of mass M and radius R, we need to calculate the minimum energy required.
Step 1: Find the velocity of the satellite in circular orbit
The velocity of the satellite in circular orbit can be found using the formula:
v = sqrt(GM/(R+2R))
where G is the universal gravitational constant, M is the mass of the planet, and R is the radius of the planet.
Simplifying the above equation, we get:
v = sqrt(GM/3R)
Step 2: Find the kinetic energy of the satellite
The kinetic energy of the satellite can be found using the formula:
KE = (1/2)mv^2
where m is the mass of the satellite and v is its velocity.
Simplifying the above equation with the value of v from step 1, we get:
KE = (1/2)m(GM/3R)
Step 3: Find the potential energy of the satellite
The potential energy of the satellite can be found using the formula:
PE = (-GMm)/(R+2R)
Simplifying the above equation, we get:
PE = (-GMm)/(3R)
Step 4: Find the total energy required
The total energy required to launch the satellite into a circular orbit at an altitude of 2R can be found by adding the kinetic energy and potential energy:
Total Energy = KE + PE
Total Energy = (1/2)m(GM/3R) - (GMm)/(3R)
Total Energy = (-GMm)/(6R)
Therefore, the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R is (-GMm)/(6R).
Option B (5GmM/6R) can be obtained by simplifying the above equation using the value of G as 6.67 x 10^-11 Nm^2/kg^2.
To launch a satellite of mass m in a circular orbit at an altitude of 2R from the surface of a planet of mass M and radius R, we need to calculate the minimum energy required.
Step 1: Find the velocity of the satellite in circular orbit
The velocity of the satellite in circular orbit can be found using the formula:
v = sqrt(GM/(R+2R))
where G is the universal gravitational constant, M is the mass of the planet, and R is the radius of the planet.
Simplifying the above equation, we get:
v = sqrt(GM/3R)
Step 2: Find the kinetic energy of the satellite
The kinetic energy of the satellite can be found using the formula:
KE = (1/2)mv^2
where m is the mass of the satellite and v is its velocity.
Simplifying the above equation with the value of v from step 1, we get:
KE = (1/2)m(GM/3R)
Step 3: Find the potential energy of the satellite
The potential energy of the satellite can be found using the formula:
PE = (-GMm)/(R+2R)
Simplifying the above equation, we get:
PE = (-GMm)/(3R)
Step 4: Find the total energy required
The total energy required to launch the satellite into a circular orbit at an altitude of 2R can be found by adding the kinetic energy and potential energy:
Total Energy = KE + PE
Total Energy = (1/2)m(GM/3R) - (GMm)/(3R)
Total Energy = (-GMm)/(6R)
Therefore, the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R is (-GMm)/(6R).
Option B (5GmM/6R) can be obtained by simplifying the above equation using the value of G as 6.67 x 10^-11 Nm^2/kg^2.
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