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Hi tmesser, thanks for the question. First, let's see if I understand why you're considering $v = \dfrac{v_f}{2}$. Does what you're thinking come from the equation $v = at$? If so, it's correct to say that after half the time, you would expect to reach half the velocity. However, that's after half the time has passed, whereas this question is asking for the velocity after half the distance has been covered. Since the car is accelerating, it will reach it's "half way velocity" at a time before the "half way distance", which you noticed with your calculation giving $37.5 \textrm{ m/s}$, whereas the velocity at the half-way position is $53 \textrm{ m/s}$.