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If you've ever sat in an accelerating car, you know that there's an invisible "force" pushing you backwards. When the car turns left, you get pushed to the right. When an elevator begins its ascent, you feel heavier.
These invisible "forces" that we feel are actually fictitious. We are so accustomed to Newton's 2nd law that we forget that it doesn't hold in non-inertial (accelerating) reference frames. With respect to the reference frame of the car, an object will indeed accelerate backwards when the car speeds up; however, there is no force that is causing it to do so. The intuition that if there is acceleration there must be a force causing it is entirely based on Newton's Second Law - so don't apply it in non-inertial frames!
To understand and calculate in these situations, look at the whole picture from an inertial reference frame and proceed with standard calculations. Remember not to include imaginary forces in free body diagrams.