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?