On a velocity-versus-time graph, which quantity equals the object's displacement?

• A. The maximum velocity
• B. The intercept with the t-axis
• C. The slope of the graph
• D. The area between the graph and the t-axis

Answer: (D)

Displacement is the net change in position, which comes from velocity being the rate at which position changes. Over a small interval, the position change is Δx ≈ v Δt. If you sum up all those tiny changes over a time interval, you’re effectively integrating velocity over time. Graphically, that sum corresponds to the area under the velocity-versus-time curve between the start and end times. If velocity stays positive, the area adds to displacement; if velocity goes negative, that area subtracts, giving the net displacement. The maximum velocity tells you how fast you’re going at a moment, the intercept with the t-axis indicates when velocity is zero, and the slope of the graph is acceleration. The area under the v–t graph is the quantity that matches the object's displacement.