Calculating the mass of matter required to accelerate a spaceship

[dylhynes]

Homework Statement

A spaceship is powered by a matter-antimatter reactor. Suppose you have a 23000-metric-ton spaceship and wish to accelerate it to 0.03 c. What is the total mass of matter and anti matter required, assuming a 100 % efficient engine.

Homework Equations

Really not sure about this question, but thinking that E=mc^2 and F=ma will be used somewhere

The Attempt at a Solution

Not sure how to approach this problem, any help would be greatly appreciated.

 

[voko]

You are right about ## E = mc^2 ##. This gives you the amount of matter/anti-matter one needs to annihilate to obtain energy ## E ##. However, ## F = ma ## is not good - it is only valid for slow motion, and 0.03 c may be too high for it. Instead, consider the relativistic kinetic energy of some mass M moving at 0.03 c.

 

[dylhynes]

Ahh, I understand what I did wrong now. I overlooked the fact that the spaceship only has kinetic energy while it is travelling. I solved for K_E, then plugged that back into E=mc². Rearranging for m I got the answer of 1.0x10⁴ kg, which was correct. Thank you very much. Also if you don't mind answering, how did you type those equations in that math font ? I am new here.

 

[voko]

Hit the quote button my message, and see the code for that. While you are there, click the ## \Sigma ## symbol at the top right of the text box, it will have a reference for much more.

 

[Nickie]

Hello,

Thank you for your question. I am happy to assist you in solving this problem.

Firstly, let's define some terms. The mass of matter refers to the amount of physical substance present in an object, while antimatter is the opposite of matter and has the same mass but opposite charge. The equation E=mc^2, also known as Einstein's famous equation, states that energy (E) is equal to mass (m) times the speed of light squared (c^2). This equation is used to calculate the amount of energy released from a mass in a nuclear reaction.

In this case, we have a spaceship with a mass of 23000 metric tons (which is equivalent to 23 million kilograms) and we want to accelerate it to 0.03 times the speed of light (c). To determine the total mass of matter and antimatter required, we can use the equation F=ma, where F is the force required to accelerate the spaceship, m is the mass of the spaceship, and a is the acceleration.

Since we know the mass (m) and the desired acceleration (a), we can rearrange the equation to solve for the force (F). This gives us F=ma = (23 million kg)(0.03c) = 6.9x10^11 Newtons.

Now, we can use this force value to calculate the total mass of matter and antimatter required. To do this, we can use the equation F=dp/dt, where dp is the change in momentum and dt is the change in time. Since we know the mass of the spaceship and the desired acceleration, we can calculate the change in momentum by multiplying the mass by the desired velocity (0.03c). This gives us dp = (23 million kg)(0.03c) = 6.9x10^8 kgm/s.

Next, we can rearrange the equation to solve for the change in time (dt). This gives us dt = dp/F = (6.9x10^8 kgm/s)/(6.9x10^11 N) = 1 second.

Therefore, we can conclude that in order to accelerate a 23000-metric-ton spaceship to 0.03c, we would need a force of 6.9x10^11 Newtons and a change in time of 1 second. Now, to determine the total mass of matter and