(a) Determine the amount of work (in joules) that must be done on a 104 kg payload to elevate it to a height of 993 km above the Earth's surface.
______ MJ
(b) Determine the amount of additional work that is required to put the payload into circular orbit at this elevation.
_____J
I have absolutley no idea how to find this problem, and it is also not in our book. If anybody has any idea, can you please help!!!
THANKS!! :)
It is in your book, the concept is energy. Potential energy is mgh. That answers a.
In b, you have to add the Kinetic energy
at orbit, centripetal force equals gravitational force.
mv^2/R= GMe*m/distance^2
1/2 mv^2= 1/2 GMe*m/distance^2
where distance is radiusEarth+ heightgiven.
Potential Energy = -GMemobject[(1/rb)-(1/ra)]
a) = -GMm[(1/2Re)-(1/Re)]
b) -GMm[(1/(Re+999000)-(1/Re)] =
X joules
X joules/1000000 = X Milijoules
c) I have no idea.... I think it should be (Orbital energy - X joules) but this is giving me the wrong answer for some reason...
Orbital Energy = -GMm/(2(Re+999000))
I get a wrong answer even if I switch signs...
(a) To determine the amount of work required to elevate the payload to a height of 993 km above the Earth's surface, we can calculate the change in potential energy.
The formula for potential energy is given by: potential energy = mass x gravitational acceleration x height.
mass = 104 kg
gravitational acceleration = 9.8 m/s^2 (approximately)
height = 993 km = 993,000 m
potential energy = 104 kg x 9.8 m/s^2 x 993,000 m = 1.01946 x 10^12 J
However, the question asks for the answer in megajoules (MJ). To convert joules to megajoules, divide the answer by 1,000,000.
1.01946 x 10^12 J ÷ 1,000,000 = 1.01946 x 10^6 MJ
Therefore, the amount of work required to elevate the payload to a height of 993 km above the Earth's surface is approximately 1.01946 x 10^6 MJ.
(b) To determine the additional work required to put the payload into circular orbit at this elevation, we need to calculate the difference between the potential energy at the given elevation and the kinetic energy required for circular motion.
The formula for kinetic energy is given by: kinetic energy = 1/2 x mass x velocity^2.
To put the payload into circular orbit, the velocity must be such that the centripetal force equals the gravitational force.
Using the formula: mv^2/R = G(M_e)(m)/distance^2, where R is the distance from the center of the Earth to the orbiting object (radius of the Earth + elevation), G is the gravitational constant, M_e is the mass of the Earth, m is the mass of the payload.
R = radius of the Earth + height = 6,371 km + 993 km = 7,364 km = 7,364,000 m (approximately)
G = 6.67408 x 10^(-11) m^3/kg/s^2 (gravitational constant)
m = 104 kg
Now we can solve for v.
(104 kg)(v^2)/(7,364,000 m) = (6.67408 x 10^(-11) m^3/kg/s^2)(5.972 x 10^24 kg)/(7,364,000 m)^2
Simplifying the equation, we find:
v^2 = (6.67408 x 10^(-11) m^3/kg/s^2)(5.972 x 10^24 kg) / (7,364,000 m)
v^2 ≈ 5.47449 x 10^7 m^2/s^2
Taking the square root of both sides:
v ≈ 7,398 m/s
Now we can calculate the kinetic energy:
kinetic energy = 1/2 x mass x velocity^2
kinetic energy = 1/2 x 104 kg x (7,398 m/s)^2
kinetic energy ≈ 3.62994 x 10^8 J
Therefore, the additional work required to put the payload into circular orbit at the given elevation is approximately 3.62994 x 10^8 J.
To determine the amount of work required to elevate the payload to a height of 993 km above the Earth's surface, you can use the concept of potential energy. The formula for potential energy is given by:
Potential Energy = mass * gravitational acceleration * height
In this case, the mass of the payload is 104 kg and the height is 993 km. The value of the gravitational acceleration can be approximated as 9.8 m/s^2.
Therefore, the potential energy can be calculated as:
Potential Energy = 104 kg * 9.8 m/s^2 * 993,000 m
To convert this to joules, you can multiply the result by 1,000, since 1 megajoule (MJ) is equal to 1,000,000 joules (J).
So, the amount of work done on the payload to elevate it to 993 km above the Earth's surface is:
Potential Energy = 104 kg * 9.8 m/s^2 * 993,000 m * 1,000 J/MJ
Simplifying this expression will give you the answer in joules or J.
To determine the amount of additional work required to put the payload into circular orbit at this elevation, you need to consider the concept of kinetic energy. At orbital height, the centripetal force exerted by the circular motion of the payload is equal to the gravitational force.
The formula to calculate the centripetal force is:
Centripetal Force = mass * velocity^2 / radius
In this case, the mass of the payload is still 104 kg. The radius of the orbit is the distance from the center of the Earth to the orbit's elevation, which is the radius of the Earth plus the given height of 993 km. The gravitational force is given by the formula:
Gravitational Force = (Gravitational Constant * Mass of Earth * mass) / radius^2
Here, the mass of the Earth is a known value, and the gravitational constant is a constant value. By setting the centripetal force equal to the gravitational force, you can solve for the velocity of the payload.
Once you have the velocity, you can calculate the kinetic energy using the formula:
Kinetic Energy = 1/2 * mass * velocity^2
This will give you the additional work required to put the payload into a circular orbit at the given elevation.
Remember to convert the result to Joules or J, if necessary, based on the units provided in the question.