When I try to derive acceleration, it turns out as a = v_f / v_i·t – why is this wrong?

The expression a = v_f / (v_i·t) is algebraically incorrect because it mishandles the basic acceleration definition a = (v_f - v_i) / t. The error likely stems from incorrectly simplifying the difference of velocities (v_f - v_i) as a division (v_f / v_i), which fundamentally changes the mathematical operation. Subtraction and division are entirely different operations: 10 - 5 = 5, but 10 / 5 = 2. Applying this to velocities, if a car accelerates from 20 m/s to 30 m/s in 5 seconds, the correct calculation is a = (30 - 20) / 5 = 2 m/s², while the incorrect formula gives (30 / 20·5) = 0.3 m/s²—an entirely different value.

This mistake commonly occurs when students confuse the fraction bar in a = Δv/Δt, thinking the slash means "divide each part" rather than "divide the difference by time." The correct interpretation is a = (change in velocity) ÷ (change in time), where "change in velocity" is a single quantity (Δv = v_f - v_i), not a ratio of velocities. Dimensionally, the incorrect formula also fails: v_f / (v_i·t) produces units of (m/s) / ((m/s)·s) = 1/s², missing the meter entirely—acceleration must have units of m/s² to be physically meaningful.

To avoid this error, always write out the full subtraction explicitly before dividing by time. Practice reading formulas aloud correctly: "acceleration equals final velocity minus initial velocity, all divided by time," emphasizing that the entire difference gets divided. When deriving from first principles, start with the definition that acceleration is the rate of velocity change, write Δv = v_f - v_i as a separate step, then form the ratio Δv/Δt = (v_f - v_i)/t. This step-by-step approach prevents algebraic shortcuts that introduce errors and reinforces the conceptual meaning behind each term.