(a) Determine the amount of work (in joules) that must be done on a 95 kg payload to elevate it to a height of 1007 km above the Earth's surface.
1 ? MJ
(b) Determine the amount of additional work that is required to put the payload into circular orbit at this elevation.
2 ? J
To determine the amount of work required to elevate the payload to a height of 1007 km above the Earth's surface, we can use the formula:
Work = Force x Distance
(a) First, we need to calculate the force required to lift the payload. This can be done using the formula:
Force = Weight = mass x gravitational acceleration
The mass of the payload is given as 95 kg.
The gravitational acceleration near the Earth's surface is approximately 9.8 m/s^2.
Force = 95 kg x 9.8 m/s^2 = 931 N
Now, let's calculate the distance:
Distance = height
Distance = 1007 km = 1007,000 m
Now, we can calculate the work:
Work = Force x Distance
Work = 931 N x 1007,000 m
Since work is measured in joules (J), the answer is in joules.
(b) To determine the additional work required to put the payload into circular orbit at this elevation, we can use the formula:
Additional Work = Total Mechanical Energy - Initial Mechanical Energy
In this case, the total mechanical energy is the sum of the gravitational potential energy and the kinetic energy when the payload is in circular orbit.
The gravitational potential energy can be calculated using the formula:
Gravitational Potential Energy = mass x gravitational acceleration x height
Gravitational Potential Energy = 95 kg x 9.8 m/s^2 x 1007,000 m
The kinetic energy when the payload is in circular orbit can be calculated using the formula:
Kinetic Energy = (1/2) x mass x velocity^2
Since the payload is in circular orbit, the velocity can be calculated using the formula:
velocity = √(gravitational constant x mass of Earth / radius)
The radius can be calculated by adding the radius of the Earth to the height above the Earth's surface:
radius = radius of Earth + height
Now, we can calculate the kinetic energy:
Kinetic Energy = (1/2) x 95 kg x (velocity)^2
The total mechanical energy is the sum of the gravitational potential energy and the kinetic energy.
Finally, we can find the additional work by subtracting the initial mechanical energy from the total mechanical energy:
Additional Work = Total Mechanical Energy - Initial Mechanical Energy
The answer will be in joules (J).