The problem involves a cannon firing bullets into a massive log, exploring the dynamics of momentum transfer and energy loss. Key questions include determining the log's velocity after successive bullet impacts, the heat energy generated, and the time intervals between impacts. The context is rooted in conservation of momentum and kinetic energy principles.
Some participants have provided calculations for the log's velocity and energy loss, while others express uncertainty about the methods used and seek clarification on specific notations and assumptions. The discussion is ongoing, with multiple interpretations being explored.
There is an assumption that the initial velocity of the log is zero. Participants are also considering the implications of the penetration time of bullets being short relative to the time interval between shots.
If you take the sum from 1 to N-1Karol said:The sum should indeed be from 2 to N since, i think, for example if i have 3 bullets then i have to add 2 intervals:
$$\sum_2^N \frac{N+99}{100}=\sum_2^N N+(N-1)99=\frac{(N-1)(N+2)}{2}+99(N-1)=\frac{(n-1)(N+200)}{200}$$
But it's strange that if i take the sum from 1 to (N-1) it gives a different thing:
$$\sum_1^{N-1} \frac{N+99}{100}=\frac{N(N+197)}{200}$$
I thought only the number of members in the sum counts, but no.