Velocity describes how position changes with time, so it is one of the most important ideas in motion. Average velocity tells you the overall rate of change across a time interval, while instantaneous velocity tells you the rate at one specific moment. On a position-time graph, both ideas are shown using slopes.
This makes graphs a powerful way to connect motion, numbers, and visual patterns.
Average velocity is found from the slope of a secant line connecting two points on the position-time curve. Instantaneous velocity is found from the slope of a tangent line at a single point, which represents what the secant slope approaches as the time interval gets smaller. If the graph is curved, the velocity is changing, so average and instantaneous velocity may not be the same.
This distinction is essential for understanding acceleration, real motion, and calculus-based physics.
Key Facts
- Average velocity: v_avg = Δx/Δt = (x2 - x1)/(t2 - t1)
- Instantaneous velocity: v = lim Δt→0 Δx/Δt
- On a position-time graph, velocity equals the slope of the graph.
- A secant line through two points gives average velocity over that interval.
- A tangent line at one point gives instantaneous velocity at that time.
- If x(t) is known, instantaneous velocity is v(t) = dx/dt.
Vocabulary
- Position
- Position is an object's location measured relative to a chosen origin.
- Average velocity
- Average velocity is the change in position divided by the change in time over an interval.
- Instantaneous velocity
- Instantaneous velocity is the velocity of an object at one specific moment in time.
- Secant line
- A secant line is a line that intersects a curve at two points and shows the average slope between them.
- Tangent line
- A tangent line is a line that touches a curve at one point and has the same slope as the curve there.
Common Mistakes to Avoid
- Using total distance instead of displacement for average velocity is wrong because velocity depends on change in position, including direction.
- Reading the height of a position-time graph as velocity is wrong because velocity is the slope of the graph, not the position value.
- Assuming average velocity equals instantaneous velocity is wrong for curved position-time graphs because the slope changes with time.
- Ignoring units is wrong because position divided by time must produce velocity units such as meters per second.
Practice Questions
- 1 A runner moves from x = 2 m at t = 1 s to x = 14 m at t = 5 s. What is the average velocity over this interval?
- 2 An object has position x(t) = 3t^2 + 2t in meters. Find its instantaneous velocity at t = 4 s.
- 3 A position-time graph curves upward and becomes steeper as time increases. Explain whether the object's velocity is increasing, decreasing, or constant, and justify your answer using slope.