If we know F(t), the force as a function of time, for straight-line motion, Newton's second law gives us a (t), the acceleration as a function of time. We can then integrate a (t) to find v (t) and x (t). However, suppose we know F( v) instead.
(a) The net force on a body moving along the x-axis equals -ev2. Use Newton's second law written as ΣF = m dv / dx and two integrations to show that x - Xo = (m/C) in (vo/v).
(b) Show that Newton's second law can be written as ΣF = mv dv/dx. Derive the same expression as in part (a) using this form of the second law and one integration.