Final answer:
To find the separation between the car and motorcycle as a function of time, we use the equations of motion for the car and motorcycle and substitute them into the equation xc - xp = xo,c - xo,p + (vc - vp)t + 1/2(ac - ap)t2.
Step-by-step explanation:
To derive an equation for the separation between the car and motorcycle as a function of time, we need to consider the motion equations of both vehicles. Let's assume the car is car 1 and the motorcycle is car 2.
The equation for the car's displacement is xc = xo,c + vct + 1/2act2, where xo,c is the initial position of the car, vc is the velocity of the car, ac is the acceleration of the car, and t is the time.
The equation for the motorcycle's displacement is xp = xo,p + vpt + 1/2apt2, where xo,p is the initial position of the motorcycle, vp is the velocity of the motorcycle, ap is the acceleration of the motorcycle, and t is the time.
The separation between the car and motorcycle is given by xc - xp, substituting the equations for their displacements we get: xc - xp = (xo,c - xo,p) + (vc - vp)t + 1/2(ac - ap)t2.