Position, Displacement & Velocity
- Distinguish position, distance, and displacement using a sign convention
- Compute displacement as a change in position, Δx = x_f − x_i
- Calculate average velocity and contrast it with average speed
Position needs an origin and a direction
Before you can describe motion you must choose a reference frame: an origin (where x = 0) and a positive direction. Position x is then a signed number — a point 4 m to the right of the origin is x = +4 m, a point to the left is x = −4 m. Every quantity in kinematics inherits that sign convention, so pick one and hold it for the whole problem.
Displacement vs. distance
Displacement is the change in position, Δx = x_f − x_i. It is a vector: it carries a sign (direction) and only cares about where you started and where you ended. Distance is the total path length traveled — a scalar that never decreases. Walk 40 m east then 15 m back west and your distance is 55 m, but your displacement is only 25 m east.
Average velocity
Average velocity is displacement divided by the time interval, so it too is a vector and can be negative. Average speed is distance divided by time and is never negative. Because displacement can be smaller than distance, average speed is always greater than or equal to the magnitude of average velocity.
At t = 2 s a cart is at x = 10 m; at t = 6 s it is at x = 34 m. Find its average velocity.
- 1.Identify the values: x_i = 10 m at t_i = 2 s, x_f = 34 m at t_f = 6 s.
- 2.Displacement: Δx = 34 m − 10 m = 24 m.
- 3.Time interval: Δt = 6 s − 2 s = 4 s.
- 4.Divide: v_avg = Δx / Δt = 24 m ÷ 4 s.
Displacement is final minus initial, not initial minus final. Flipping that subtraction is the most common Unit 1 sign error and reverses the direction of your answer.
A runner jogs 40 m east, then turns around and jogs 15 m west. What is the runner’s displacement?
A swimmer swims 50 m to the far wall of a pool and then straight back to the start, taking 40 s in total. What is her average velocity for the round trip?
A car’s position changes from x = +8 m to x = −4 m in 3 s. What is its average velocity?
On the AP exam, always state your positive direction before plugging in numbers. Half of kinematics is bookkeeping — a consistent sign convention earns points that raw algebra cannot.
Answer the 3 checkpoints as you read.
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