How Does Mass Conversion Impact the Dynamics of a Relativistic Spaceship?

In summary, the velocity of the spaceship at time t is given by v=\frac{\sqrt{2rmt-r^2t^2}}{m}c, its acceleration is given by a=\frac{rm-r^2t}{m\sqrt{2rmt-r^2t^2}}c, and the distance it has traveled at time t is \displaystyle\int^{t}_{0} \frac{\sqrt{2rmx-r^2x^2}}{m}c\, dx. The distance it has traveled just before it becomes purely energy is not defined in this system, as the mass will decrease exponentially. The time elapsed with respect to the
  • #1
lotrgreengrapes7926
32
0
A spaceship of mass [tex]m\ \text{kg}[/tex] is propelled by converting [tex]r\ \text{kg}[/tex] of its mass into energy every second. Assume no friction and a perfectly efficient system.

1) Find the velocity of the spaceship at time [tex]t[/tex].

2) Find its acceleration at time [tex]t[/tex].

3a) Find the distance it has traveled at time [tex]t[/tex].
3b) Find the distance it has traveled just before it becomes purely energy.

4a) Find how much time has elapsed with respect to the spaceship at time [tex]t[/tex].
4b) Find how much time has elapsed with respect to the spaceship by the time it becomes purely energy.

This isn't homework, but I just wanted to check that my methods were correct.
 
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  • #2
What methods did you use?

You obviously want to use conservation of energy and momentum. I don't know if I understand 3b) and 4b) because under your proposed system the mass will decrease exponentially, so it will take inifinite time and what time is just before infinity?
 
  • #3
The amount of matter being converted to energy is not proportional to the mass, it is constant. For example, for m=1000 and r=1, it will take 1000 seconds.
1) [tex]v=\frac{\sqrt{2rmt-r^2t^2}}{m}c[/tex]
2) [tex]a=\frac{rm-r^2t}{m\sqrt{2rmt-r^2t^2}}c[/tex]
3a) [tex]\displaystyle\int^{t}_{0} \frac{\sqrt{2rmx-r^2x^2}}{m}c\, dx[/tex]
I don’t know how to integrate that.
3b) ??
I’m not sure about number 4.
4a) [tex]t^2-\frac{r}{2m}t^3[/tex]
4b) [tex]\frac{m^2}{6r^2}[/tex]
 

Related to How Does Mass Conversion Impact the Dynamics of a Relativistic Spaceship?

What is a "Relativistic Spaceship"?

A "Relativistic Spaceship" refers to a hypothetical spacecraft that travels at extremely high speeds, approaching the speed of light. This concept is based on Einstein's theory of relativity, which suggests that objects traveling at such speeds experience time dilation and other strange effects.

What is time dilation?

Time dilation is a phenomenon that occurs when an object moves at high speeds, causing time to pass differently for the object compared to a stationary observer. This effect becomes more significant as the object approaches the speed of light, with time appearing to slow down for the moving object.

Can a spaceship really travel at the speed of light?

According to Einstein's theory of relativity, the speed of light is the maximum speed possible in the universe. While it is not currently possible for a spaceship to reach this speed, scientists are constantly researching and developing ways to approach this limit, such as using antimatter propulsion or harnessing the energy of black holes.

What other effects can occur on a relativistic spaceship?

Besides time dilation, other effects that can occur on a relativistic spaceship include length contraction, where an object's length appears to shorten in the direction of motion, and mass dilation, where an object's mass appears to increase as it approaches the speed of light. These effects can also lead to strange phenomena such as the "twin paradox", where one twin ages significantly slower than the other due to traveling at high speeds.

Why is studying relativistic spaceships important?

Studying relativistic spaceships is important because it helps us understand the fundamental laws of the universe and how they can be applied to technology. It also has practical applications in space exploration, as it can provide insights into how we can travel to distant planets and galaxies at high speeds, and how we can potentially overcome the limitations of our current propulsion systems.

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