Car shakes badly when accelerating anywhere between 35 and 50mph on 1999 Volkswagen Golf

My car shakes pretty violently anytime I am accelerating and get close to the 40mph mark. It seems to be less violent after I exceed 50mph and seems to only happen when I am accelerating. What could this be?

Asked by for the 1999 Volkswagen Golf
If the car's engine runs and idles smoothly then I would look at your inner CV joints.