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I'm not sure I get why this makes sense. Okay, it takes 4 times as much energy to accelerate an object twice as much cause it is a squared function e=(1/2)mv^2 however when objects collide, objects moving twice as fast don't affect the other objects 4 times as much so what gives?

I'm not sure I get why this makes sense. Okay, it takes 4 times as much energy to accelerate an object twice as much cause it is a squared function e=(1/2)mv^2 however when objects collide, objects moving twice as fast don't affect the other objects 4 times as much so what gives? for example if you accelerated a 4 pound bowling ball to 2mph and then threw a 16 pound bowling ball at 1mph the opposite direction the sixteen pound bowling ball would over power the 4 pound because it has more momentum p=mv and momentum is conserved, but the amount of energy it took accelerating the two would be the same, so why doesn't this violate conservation of energy?

What do you mean by “effect”? If you mean “causes damage” then you’re wrong. The amount of damage caused to one body by another is determined by the other body’s kinetic energy, not its momentum. It’s change in motion is determined by its momentum.