A vehicle of mass \(m\) starts from rest at time \(t = 0\) and accelerates along a straight, horizontal track. An engine delivers an instantaneous mechanical power output \(P(v)\) that varies with the vehicle's speed \(v\) according to the relation \(P(v) = P_0\left(1 - \dfrac{v}{v_0}\right)\), where \(P_0\) and \(v_0\) are positive constants, for speeds \(0 \le v < v_0\). Assuming resistive forces are negligible, which of the following expressions represents the time \(t_1\) required for the vehicle to reach a speed of \(\dfrac{v_0}{2}\)?