# What would be different if forces gave rise to velocity or jerk?

The way I understand it, force gives rise to acceleration. The integral of acceleration is velocity, and the integral of velocity is position. What if forces gave rise to velocity directly? Or what if they gave rise to jerk (the derivative of acceleration)?

In the $F=mv$ world, momentum and force are equivalent. Push an object hard, and it will instantly begin to move. If gravity acts as a force (it could still be an acceleration if we want a slightly different scenario) then everything close to a mass will move towards it, imploding towards higher densities until the pressure balances gravity. There would not be any orbits since velocities would be pointed straight at the main mass. A dropped ball would follow the equation $x'(t)=-g$, that is, $x(t)=-gt + x(0)$: objects fall with constant speed - in fact, when you throw a ball, as soon as it leaves your hand it will change speed to gravity speed.
In the $F=mj$ world, accelerations would be much softer. You would need to press something longer to make it move. A thrown ball would follow the equation $x'''(t)=-g$, that is, $x(t)=-(1/6)g t^3 + a_0 t^2 + v_0t + x(0)$: if released from rest it will take longer for it to drop but the speed will grow even faster. Presumably it would also keep a trace of the acceleration you gave it: throw it hard upward, and it will not just rise but keep on accelerating for a while until gravity won out.
• Does the $F=mj$ world have orbits? Does it have any other stable dynamic systems? – Neil G Dec 18 '17 at 12:50