Here is a made up function of force changing with respect to displacement: F(x) = 2000 - 100x The force is applied onto an object, pushing it horizontally. Ignore friction and air resistance. The mass is constant. I'm trying to find Velocity with respect to displacement. Attempt 1: Knowing F = m*a: 2000 - 100x = m*a a(x) = (2000-100x)/m To find velocity, I take the integral of a(x) to get v(x) = [-100(0.5x2 - 20x)]/m Attempt 2: Now, I believe that Work = Force * Displacement. W = ∫ from 0 to b of (2000 - 100x)dx = [-50(b-40)b] Since the object is only sliding horizontally, there's no potential energy, but there is kinetic energy. I set [-100(0.5x2 - 20x)] = 0.5mv^2 and solve for v: v(x) = sqrt[(2/m) * [-50(x-40)x]] In the end, I get two different answers. What I would like to know is what am I doing wrong? Did I use illegal math? Did I get the formulas wrong? Did I use the forumlas wrong?