Serway & Jewett — Circular Motion and Other Applications of Newton's Laws: Problem 46

For \( t < 0 \), an object of mass \( m \) experiences no force and moves in the positive \( x \) direction with a constant speed \( v_i \). Beginning at \( t = 0 \), when the object passes position \( x = 0 \), it experiences a net resistive force proportional to the square of its speed: \( \vec{F}_{net} = -m k v^2 \hat{i} \), where \( k \) is a constant. The speed of the object after \( t = 0 \) is given by \( v = v_i / (1 + k v_i t) \). (a) Find the position \( x \) of the object as a function of time. (b) Find the object’s velocity as a function of position.