The slope of the tangent to a position-time graph equals which quantity?

• A. Speed
• B. Average velocity
• C. Acceleration
• D. Instantaneous velocity

Answer: (D)

On a position-time graph, the horizontal axis is time and the vertical axis is position. The slope of the curve at any moment tells you how quickly position is changing at that exact moment. That rate of change is velocity, and using the tangent line at that specific time gives the instantaneous velocity—the velocity right at that instant. If the tangent slopes upward, the instantaneous velocity is positive (moving in the positive direction); if it slopes downward, it’s negative (moving in the opposite direction). This is different from average velocity, which would use the slope of a line connecting two points on the graph, and from acceleration, which is how quickly the velocity itself changes over time. Speed is the magnitude of velocity, so it’s not the slope itself unless you ignore direction. Therefore, the slope of the tangent to a position-time graph equals the instantaneous velocity.