How much work is done when a satellite is launched into orbit?

In summary: Why don’t you try and see what you get?I can try and get the equation for kinetic energy, but I'm not sure if it will give me the correct answer.
  • #1
tuki
19
1

Homework Statement


A Satellite is brought up into a geostationary orbit (altitude 35800km measured from the surface of the earth). Satellite weights 1000.0kg. How much work is required to bring satellite from a surface of the Earth to
geostationary orbit?

Homework Equations


Newton's law of universal gravitation
$$ F = G \frac{m_1m_2}{r^2} $$
Work is defined as:
$$ W \int F \cdot s ds $$

The Attempt at a Solution


I want to compute work done by Newton's law of universal gravitation when
moving satellite from the surface of the Earth to the orbit. It should be done with:h0 is surface of the Earth (about 6371 km)
h1 is geostationary orbit (about 6371 km + 35800 km = 42171km)
m1 is mass of the satellite (about 1000 kg)
m2 is mass of the Earth (about 5.972E24)

$$ W = \int_{h_0}^{h_1} G \frac{m_1 m_2}{r^2} dr = - G \frac{m_1 m_2}{h_1} - (- G \frac{m_1 m_2}{h_0} $$
$$ = G m_1 m_2 (\frac{1}{h_1} - \frac{1}{h_0}) \approx 5.3108\cdot 10^{10} \text{ J} $$

However, our textbook suggests that the correct answer would be 5.77E10 Joules. I can't exactly spot out where the mistake is.
 
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  • #2
Is giving the satellite potential energy the only thing you need to do to put it in geostationary orbit?
 
  • #3
Orodruin said:
Is giving the satellite potential energy the only thing you need to do to put it in geostationary orbit?
Yes you would need the velocity too in order to stay in orbit.
 
  • #4
Can you see that that's your answer, or is still unclear?
 
  • #5
Bandersnatch said:
Can you see that that's your answer, or is still unclear?
Yes, I can get the correct answer by adding kinetic energy from the speed $$ E_{kinetic} = \frac{1}{2} mv^2 $$ to the potential energy I already computed. I wonder if it would be possible to derive the work required to accelerate an object to a given speed (when mass is known) starting from Newton's second law? $$ F = ma $$ I mean you start from Newton's second law and end up with a formula for kinetic energy?
 
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  • #6
tuki said:
I wonder if it would be possible to derive the work required to accelerate an object to a given speed (when mass is known) starting from Newton's second law?
F=maF=ma​
F = ma I mean you start from Newton's second law and end up with a formula for kinetic energy?
Why don’t you try and see what you get?
 

Related to How much work is done when a satellite is launched into orbit?

1. How is the amount of work calculated when a satellite is launched into orbit?

The amount of work done when launching a satellite into orbit is calculated by multiplying the force exerted on the satellite by the distance it is moved. This force is typically provided by the rocket engines and the distance is the height of the satellite's orbit.

2. What factors contribute to the amount of work required to launch a satellite into orbit?

The amount of work required to launch a satellite into orbit is affected by various factors including the weight of the satellite, the distance it needs to travel, and the force of gravity acting on the satellite. Other factors such as atmospheric conditions and the efficiency of the rocket also play a role.

3. How does the type of orbit affect the amount of work needed to launch a satellite?

The type of orbit chosen for a satellite can greatly impact the amount of work required for its launch. For example, a low Earth orbit requires less work compared to a geostationary orbit, as the satellite needs to travel a shorter distance and the force of gravity is weaker at lower altitudes.

4. Is the amount of work done constant throughout the entire process of launching a satellite into orbit?

No, the amount of work done during the launch of a satellite is not constant. The amount of work required is highest during the initial stages of launch when the rocket engines are providing the most force, and decreases as the satellite gains altitude and reaches its designated orbit.

5. How does launching a satellite into orbit compare to other forms of work in terms of energy and resources?

Launching a satellite into orbit requires a significant amount of energy and resources, making it a complex and expensive process. It is often compared to other forms of work such as building a skyscraper or constructing a bridge, as they all involve overcoming gravity and require precise calculations and engineering.

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