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If there is a train moving at 2 m/s with a cannon attached to it loaded with a ball, and then the ball is launched from the train, how would one find the kinetic energy produced by the explosion if given the final speed of the ball and the final speed of the train is 0 m/s ?

Note the question specifically wants the kinetic energy produced by the explosion not the net change in kinetic energy in the system; the net change in kinetic energy in the system wouldn't necessarily correspond to this would it? The net change in KE of the system would be the change in the ball's KE minus the train's original KE. Would it make sense for the original KE of the train to be subtracted though? Wouldn't this be the explosion "taking" KE away from the train? Also, by finding the net change in ME of the system (in this case only KE changes), this gives the net work done by the non-conservative force (the explosion) but wouldn't the explosion be both adding and dissipating energy from the system (adding KE but dissipating energy as heat and internal energy). Hence the net change in KE of the system would give both the KE produced by the explosion AND the energy lost through the explosion. I think this is different from the situation where you have a stationary object and then it is exploded and then you find the sum of the KE of each of the pieces to find the KE of each piece in order to find the total KE produced by the explosion; in this case no KE is "taken away" from an object like the way that the train's KE is reduced. Could someone please help clarify this if it is a misunderstanding?

Could the KE added by the explosion be found purely through the change in kinetic energy of the ball? Would this mean that the decrease in the train's KE is caused by the dissipation of energy as internal energy and sound?

IF instead the KE added by the explosion is found through the net change in the ball+train KE (as a system), where does the KE of the train go (intuitively I can understand that if the explosion force acts in the opposite direction to its velocity then it will do work on the train, decelerating it. However, then this would still be KE produced by the explosion no? So wouldn't this have to be added to the KE produced by the explosion?). I think my main difficulty is in separating the KE produced by the explosion from the change in KE of the system (even though the KE of the train may decrease, this is partly due to the KE produced in the explosion so it should be added...?).

If someone could help clarify this, I would really appreciate it.

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2 Answers 2

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This is really about quite a simple balance of energies. Assuming noise and other non-conservative energies (like friction in the barrel) associated with the shot are negligible, then the energy balance is given by:

$$E_{k,1}+E_c=E_{k,2}$$

where $E_c$ is the chemical energy released by the shot, and the $E_k$ the kinetic energies before and after the explosion ($1$ and $2$).

So we have:

$$\frac12m_tv_t^2+E_c=0+\frac12m_bv_b^2$$

$$\boxed{E_c=\frac12(m_bv_b^2-m_tv_t^2)}$$

where the indices $t$ and $b$ refer to the train and cannon ball respectively.

So $E_c$ is the kinetic energy the explosion added to the system.

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  • $\begingroup$ This assumes that all of the change in kinetic energy of the system is due to the chemical energy of the system but is that necessarily true? For example, in an inelastic collision of two balls, kinetic energy is lost even though there is no net external force acting on them. So in this case, would it be possible for the change in energy to also be due to other factors? Would it be reasonable to assume no internal energy/sound energy lost? $\endgroup$
    – planckton
    Commented Feb 24, 2021 at 15:20
  • $\begingroup$ From your question I concluded that you're overthinking this a bit. Where would other 'factors' come from? There's a train, a cannon and some explosive, nothing else. Set up the balance of energies and that's basically it. As regards the loss of non-conservative energies, these cannot be calculated but that doesn't really matter: the calculation in my answer gives you the amount of chemical energy that wasn't lost and that is what really matters. These energies lost are in any case very small compared to the energy needed to propel the canonball. $\endgroup$
    – Gert
    Commented Feb 24, 2021 at 15:27
  • $\begingroup$ This assumes that all of the change in kinetic energy of the system is due to the chemical energy of the system but is that necessarily true? So yes: this is necessarily true unless you want magically poof! some energy into existence. $\endgroup$
    – Gert
    Commented Feb 24, 2021 at 15:37
  • $\begingroup$ Is it possible that the explosion both adds and takes away kinetic energy from the system (for example, giving the ball KE but since the explosion recoil opposes the motion of the train, it 'takes away' KE from the train)? Thus finding the change in KE of the system gives the net work/energy added into the system by the explosion not the total amount of KE provided by the explosion (this distinction is important because the next part involves calculating the amount of fuel required to produce such an explosion so the total energy released not the net energy change of the system is required) $\endgroup$
    – planckton
    Commented Feb 25, 2021 at 13:12
  • $\begingroup$ Still overthinking, I see. By all means try that approach: you'll find you won't be able to make it work. You're 'mystifying' something very basic, for no good reason. Good luck and have a nice day! $\endgroup$
    – Gert
    Commented Feb 25, 2021 at 14:50
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The work energy theorem for a system tells that the change in kinetic energy of the system is equal to the net work doe by all forces, internal and external. In this case, the external forces are neglected and the internal foces are the forces exerted by the expanding gas on the projctile and the cannon. So, what one can say for sure is that the net work of the forces exterted by the gas is equal to the change in KE of the system, which is the difference in total KE before and after. If this work is equal to the energy released by the explosion is not so obvious. The explosion does not actually "produce KE". It increases in a short period the pressure and temperature of the material in the space where the gunpowder was. Then this hot gas expands and cools down as it does work. If we consider this expansion adiabatic then the work done is equal to the change in the internal energy of the gas. But is the final internal energy equal to the inetrnal energy before the explosion? Definitely not in practice, as the gas keeps expanding and cooling after the projectile left the barrel. I think using the term "kinetic energy produced by the explosion" is not the best idea.

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