Question about conservation of energy/mass

In summary, the conversation discusses the concept of conservation of energy and how it applies to a spaceship accelerating and decelerating in outer space. The initial acceleration increases the spaceship's kinetic energy, while the deceleration decreases it. However, the expelled fuel also gains kinetic energy, balancing out the energy loss of the spaceship and demonstrating the conservation of energy.
  • #1
J Goodrich
17
0
I've been thinking about this problem today. I tried reasoning my way through it and I haven't been able to. I might be totally missing something obvious, so if so please feel free to laugh and point it out, but for me right now this is confusing. It deals with the conservation of energy/mass:

Let's say that a 10,000 kg spaceship is at rest in outer space. The crew decides that they want to then begin moving, so they fire their engines. They accelerate at 500 m/s^2 for 10 seconds.

F = mA
F = (10,000 kg)(500 m/s^2)
F = 5,000,000 N

xt = x0 + v0*t + (A*t^2)/2
xt = 0 + 0 + ((500 m)*(10 m/s^2)^2)/2
xt = 25,000 m

W = Fd
W = (5,000,000 N)(25,000 m)
W = 1.25 x 10^11 J

Their engines expended 1.25 x 10^11 joules of energy doing this process (I'm assuming 100% efficiency for simplicity, but even if it weren't it shouldn't change the nature of my problem).

vt = v0 + A*t
vt = 0 + (500 m/s^2)(10 s)
vt = 5,000 m/s

KE = (m*v^2)/2
KE = ((10,000 kg)(5,000 m/s)^2)/2
KE = 1.25 x 10^11 J

Since there was 100% efficiency, it makes sense that the energy that the ship used would then be converted to its kinetic energy. This is conservation of energy, in that the energy stored as fuel is now the ship's kinetic energy.

Now the crew of the ship decides they want to stop moving after reaching their destination. The ship then chooses to fire their rockets in the same fashion, except the opposite direction: 500 m/s^2 for 10 seconds.

The force applied to the ship by the engines is negative (because the acceleration is in the opposite direction as the burst tomorrow) and the work done on the ship is negative. Therefore, after the 10 seconds the ship's velocity will be 0 and therefore kinetic energy will be 0. However, during the burn, the engines were still expending energy as they were fired 1.25 x 10^11 J.

One would think that because the ship is reducing its kinetic energy, the engines would actually be taking that energy back in and storing it for use later. However, in this example, the engines are actually expending more energy.

So now to conclude, the ship now has 0 joules of KE and over the process has expended 2.5 x 10^11 joules of energy. What has happened to this energy from a standpoint of conservation of energy? Where is it? I don't understand how it was conserved, because although in the first half it would had been transferred to KE, once the ship decelerated I see it was being "lost forever".

What am I missing?
 
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  • #2
You need to think about what the 'engines' are doing.

Consider the case of some kind of thrusters. These work by converting stored chemical potential energy in the fuel into kinetic energy.

Acceleration
The fuel is ignited in the thrusters, causing the fuel to be expelled from the thrusters and the spaceship to be propelled forwards (conservation of momentum/Newton's 3rd law). This increases the kinetic energy of the fuel and of the spaceship.

Deceleration
The fuel is ignited in the thrusters and directed such that it is expelled outwards in the direction of travel. This causes a backward acceleration (i.e. deceleration) of the spaceship (again conservation of momentum/Newton's 3rd law). This increases the kinetic energy of the fuel and decreases the kinetic energy of the spaceship. However this time the fuel has more kinetic energy (because the spaceship was already traveling in the direction it was expelled), which balances the loss in kinetic energy of the spaceship. Energy remains conserved.

Of course in reality the expelled fuel has mass, so you would need to use the full version of Newton's second law... but that's a separate issue :smile:
 
  • #3
Ah, that makes sense. Thanks!
 

1. What is the law of conservation of energy/mass?

The law of conservation of energy/mass states that energy/mass cannot be created or destroyed, but can only be transformed from one form to another. This means that the total amount of energy/mass in a closed system remains constant over time.

2. How does the law of conservation of energy/mass apply in everyday life?

The law of conservation of energy/mass applies in many ways in our daily lives. For example, when we turn on a light bulb, electrical energy is converted into light energy. Similarly, when we eat food, the chemical energy in the food is converted into the energy our bodies need to function. In both cases, the total amount of energy/mass remains the same.

3. Are there any exceptions to the law of conservation of energy/mass?

The law of conservation of energy/mass is a fundamental law of physics and has been observed to hold true in all known cases. However, there are some cases where it may seem like energy/mass is being created or destroyed, but in reality, it is just being transformed into a different form that is not easily measurable.

4. How is the law of conservation of energy/mass related to the concept of entropy?

The law of conservation of energy/mass is closely related to the concept of entropy, which is a measure of the disorder or randomness in a system. In a closed system, energy/mass will always flow from areas of higher concentration to areas of lower concentration, resulting in an increase in entropy. This means that although the total amount of energy/mass remains constant, it becomes less organized and more dispersed over time.

5. How is the law of conservation of energy/mass applied in the field of renewable energy?

In the field of renewable energy, the law of conservation of energy/mass is applied in the design and implementation of systems that convert energy from renewable sources such as solar, wind, or hydropower into usable forms. These systems must ensure that the total amount of energy/mass remains constant and that there is no net loss of energy/mass during the conversion process.

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