# Physics question by sci-fi writer

I hope someone can help me with this. i have a question about kinetic energy and kamikazi spaceships. If i have say a 40,000 Metric ton spacecraft that rams another ship at a specified velocity the other ship not moving i understand that i would apply the kinetic energy of the ramming ship to the victim ship to estimate the damage, but how do i figure the destructive energy applied to the ramming ship?

Well Kinetic Energy is one-half the mass of the object times the square of its velocity. If it rammed a stationary object the energy imparted to the object being hit would depend on the resulting scenario.

Realistically it is impossible that the ship being hit does not move, this violates the conservation of momentum. If you want to find out the actual result of two objects of given masses colliding you can do some quick, simple kinematics to find the resulting energy transfers. If you are not worried about specifics, I would start with the kinetic energy of the colliding ship and go up from there, depending on the amount of recoil you want the projectile-esque ship to undergo.

@Vorde, I don't think he's asking about an elastic collision. He is asking that if we were to take the energy of the plane and convert it into a destructive form (explosion, bending the metal, etc.) then how can we estimate the destructive force of the plane impacting the ship?

I think typically you'll want to look into the material strength of the ship in question, will it even be damaged by the plane? What sort of force is required to deform the material? What is the melting point of the material (the fuel will most likely be ignited!!)? I think if you research these things you'll have a good idea the amount of damage that will be caused. Look into a property called Impulse in physics, this will give you a sense of how much force will be exerted on the ship when rammed.

The likely result is that the ramming ship and the ship it rammed stick together in an inelastic collision. Some energy is necessarily lost in such a collision, and the combination of the two ships stuck together will have some velocity. In fact, that energy loss is the energy spent in damaging both ships' internal structures--it's the conversion of velocity and momentum into crumpling and denting and shearing the ship itself.

Say ship 1 with mass $m_1$ is ramming ship two with mass $m_2$ and does so with speed $v$. Conservation of momentum tells us that, in a totally inelastic collision where both the ships stick together, the velocity $v'$ after collision is

$$v' = \frac{m_1 v}{m_1 + m_2} = \frac{\mu}{m_2} v$$

where $\mu = m_1 m_2/(m_1 + m_2)$ is called the "reduced mass" of the system.

The kinetic energy lost is

$$\Delta E = \frac{1}{2} (m_1 + m_2) (v')^2 - \frac{1}{2} m_1 v^2 = -\frac{1}{2} \mu v^2 < 0$$

For non-relativistic velocities, this should give a good ballpark figure; the totally inelastic collision assumption is probably only roughly good, and strictly speaking, some of the energy goes into damaging the ramming ship. Take this estimate as an upper limit on the work done damaging the target ship. As has been pointed out, just because this is the energy lost doesn't mean it gives a great idea of what kind of damage results. That energy can be lost in physically deforming the framework of the ship or in melting materials and so on.

thank you Muphrid, you are very helpful. you guys on this site are awesome. I can't wait to apply this equation, thanks.

Muphrid, i'm sorry i think i didn't say exactly what i'm after so i'll ask some questions and hope you understand. First, the equation you gave me; is it for total damage to the system(by that i mean both ships?) you see what i want to understand is that if one ship is ramming another does its position as "ramming ship" grace it with less damage than the "rammee" or does inertia demand that both ships (by ships i mean spaceships,[i must specify i just get used to talking to other sci-fi writers and they take it as a given when i say ship i'm talking spaceships]) take equal damage? if the advantage favors the ramming ship then that's what i'm after, if both take equal damage and that's what your equation gives me then i guess i'm done. you see because what i was doing is treating the "rammer" as a kinetic projectile and applying its kinetic energy to the "rammee" but wanted to know what damage the "rammer" would take assuming that value would be less. you see i don't want to go all the way down to the quantum level with this i'm just looking for something general that tells me how much energy is directed the "rammee" and how much to the "rammer". thanks in advance for all the help!

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That is the total damage for the entire system. In space, it makes no difference between one ship stationary and the other moving vs. the opposite. I won't say both take equal damage because I can't be sure of that; it really depends on materials and such. In addition, that energy can be used to overcome barriers and release even more energy--in plain language, the collision could do everything you'd expect like break open pressure vessels or ignite fuel, which would add to damage resulting from the collision.

O.K. great Muphrid thanks again. I kind of thought the damage would be equal but had to be sure thanks.