Answer :
Final answer:
The amount of work required to elevate the 101 kg payload to a height of 998 km above the Earth's surface is approximately 9.9 x 10^11 Joules. The amount of additional work required to put the payload into a circular orbit at this elevation is approximately 4.95 x 10^9 Joules.
Explanation:
To determine the amount of work required to elevate the payload to a height of 998 km above the Earth's surface, we can use the formula for gravitational potential energy:
Gravitational Potential Energy (PE) = mgh
where m is the mass of the payload, g is the acceleration due to gravity, and h is the height above the Earth's surface.
Given:
- Mass of the payload (m) = 101 kg
- Height above the Earth's surface (h) = 998 km = 998,000 m
- Acceleration due to gravity (g) = 9.8 m/s^2
Substituting the given values into the formula, we get:
PE = (101 kg)(9.8 m/s^2)(998,000 m)
Calculating the value, we find that the gravitational potential energy is approximately 9.9 x 10^11 Joules.
Therefore, the amount of work required to elevate the payload to a height of 998 km above the Earth's surface is approximately 9.9 x 10^11 Joules.
To determine the amount of additional work required to put the payload into a circular orbit at this elevation, we need to consider the concept of circular motion. In circular motion, an object experiences a centripetal force that continuously changes its direction. The work done to maintain circular motion is equal to the change in kinetic energy of the object.
Given that the payload is already at the desired elevation, the additional work required to put it into a circular orbit is equal to the change in kinetic energy. Since the payload is initially at rest, the change in kinetic energy is equal to the final kinetic energy.
The formula for kinetic energy is:
Kinetic Energy (KE) = (1/2)mv^2
where m is the mass of the payload and v is the velocity of the payload in circular orbit.
Since the payload is in a circular orbit, the velocity can be calculated using the formula for centripetal acceleration:
Centripetal Acceleration (a) = v^2/r
where r is the radius of the circular orbit.
Given:
- Mass of the payload (m) = 101 kg
- Radius of the circular orbit (r) = height above the Earth's surface (h) = 998 km = 998,000 m
Substituting the given values into the formula for centripetal acceleration, we get:
a = v^2/(998,000 m)
Since the centripetal acceleration is equal to the acceleration due to gravity (g), we can equate the two:
g = v^2/(998,000 m)
Solving for v, we find:
v = sqrt(g * 998,000 m)
Substituting the value of g (9.8 m/s^2) into the equation, we get:
v = sqrt(9.8 m/s^2 * 998,000 m)
Calculating the value, we find that the velocity of the payload in circular orbit is approximately 9,900 m/s.
Substituting the values of mass (m) and velocity (v) into the formula for kinetic energy, we get:
KE = (1/2)(101 kg)(9,900 m/s)^2
Calculating the value, we find that the kinetic energy is approximately 4.95 x 10^9 Joules.
Therefore, the amount of additional work required to put the payload into a circular orbit at a height of 998 km above the Earth's surface is approximately 4.95 x 10^9 Joules.
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Final answer:
The amount of work required to elevate the 101 kg payload to a height of 998 km above the Earth's surface is approximately 9.9 x 10^11 Joules. The amount of additional work required to put the payload into a circular orbit at this elevation is approximately 4.99 x 10^8 Joules.
Explanation:
To determine the amount of work required to elevate the payload to a height of 998 km above the Earth's surface, we can use the formula for gravitational potential energy:
Gravitational Potential Energy (PE) = mgh
where m is the mass of the payload, g is the acceleration due to gravity, and h is the height above the Earth's surface.
Given:
- Mass of the payload (m) = 101 kg
- Height above the Earth's surface (h) = 998 km = 998,000 m
- Acceleration due to gravity (g) = 9.8 m/s^2
Substituting the given values into the formula, we get:
PE = (101 kg)(9.8 m/s^2)(998,000 m)
Calculating the value, we find that the gravitational potential energy is approximately 9.9 x 10^11 Joules.
Therefore, the amount of work required to elevate the payload to a height of 998 km above the Earth's surface is approximately 9.9 x 10^11 Joules.
To determine the amount of additional work required to put the payload into a circular orbit at this elevation, we need to consider the concept of circular motion. In circular motion, an object experiences a centripetal force that continuously changes its direction. The work done to maintain circular motion is equal to the change in kinetic energy of the object.
Given that the payload is already at the desired elevation, the additional work required to put it into a circular orbit is equal to the change in kinetic energy. Since the payload is initially at rest, the change in kinetic energy is equal to the final kinetic energy.
The formula for kinetic energy is:
Kinetic Energy (KE) = (1/2)mv^2
where m is the mass of the payload and v is the velocity of the payload in circular orbit.
Since the payload is in a circular orbit, the velocity can be calculated using the formula for centripetal acceleration:
Centripetal Acceleration (a) = v^2/r
where r is the radius of the circular orbit.
Given:
- Mass of the payload (m) = 101 kg
- Radius of the circular orbit (r) = 998 km = 998,000 m
Substituting the given values into the formula for centripetal acceleration, we get:
a = v^2/(998,000 m)
Since the acceleration due to gravity (g) is equal to the centripetal acceleration, we can equate the two:
g = v^2/(998,000 m)
Solving for v, we find:
v = sqrt(g * 998,000 m)
Substituting the value of g (9.8 m/s^2) into the equation, we get:
v = sqrt(9.8 m/s^2 * 998,000 m)
Calculating the value, we find that the velocity of the payload in circular orbit is approximately 3,140 m/s.
Substituting the values of mass (m) and velocity (v) into the formula for kinetic energy, we get:
KE = (1/2)(101 kg)(3,140 m/s)^2
Calculating the value, we find that the kinetic energy is approximately 4.99 x 10^8 Joules.
Therefore, the amount of additional work required to put the payload into a circular orbit at a height of 998 km above the Earth's surface is approximately 4.99 x 10^8 Joules.
Learn more about work and energy in physics here:
https://brainly.com/question/33726554
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