kE=mv /2 : Let's say we have an object moving in the +ve x direction with mass m and a speed v and energy E (which we dont know yet). Now let's say we apply a constant force of 1N against the object (-x direction) until it stops. Then F=ma=-1. Integrate this a few times to find the distance it takes for it to stop: mv=-t+C. We know at t=0, the initial velocity of the mass was v , so C=mv . We want v=0 (when the object stops), so t=C=mv . Integrate again you get md=-1/2t +mv t, plug in t=mv : md=-1/2(mv ) +(mv ) =1/2(mv ) , so d=1/(2m)(mv ) =1/2mv . By the definition of work used before, W = F (Integral) d r = Fd=(1)(1/2)mv . So if it took us 1/2mv units of energy to stop the object, it (by conservation of energy) but have had energy of 1/2mv . This derivation makes sense, although it may not be the standard way of doing it since I just made it up... F=Kq1q2/d^2: This comes straight from Maxwell's equations (Gauss's Law) applied to a sphere or a point. I'm not gonna try to do it here just because (as I found out from the above derivation), doing math with a keyboard really sucks. What you do is just use Gauss's law on a sphere or point of charge q and find the E fiend around the object. It gets pretty easy since, because of spherical symmetry, you know the field must point radially it or out of the sphere. F=GMm/r^2: Again, (although I havent done this one), I'm pretty sure it's just Gauss's Law but applied to a gravity field. Note that both F=GMm/r^2 and F=Kq1q2/d^2 are examples of the "inverse square law", which, in general, described the intensity of something coming from a point particle (or sphere) with respect to distance. (Note that 1/r^2 term is *always* there) ለዚህ ነው man😁😂😂😂