πPosition, Displacement & Velocity
What is Position?
Position tells us where an object is, measured from a reference point called the origin. We use a coordinate system (usually x for horizontal, y for vertical) to describe position with a number and a direction.
For example, if you start at your front door and walk 5 meters to the right, your position is x = +5 m. If you then walk 3 meters back to the left, your position is x = +2 m.
Displacement is the change in position:
Ξx = x_final β x_initial
Note: Displacement is a vector β it has both magnitude and direction. Distance, on the other hand, is a scalar (just a number). If you walked 5 m right then 3 m left, your distance traveled is 8 m but your displacement is +2 m.
Think About It
A runner completes one full lap of a 400-meter track. What is their total distance traveled? What is their displacement?
Velocity vs. Speed
Average velocity is displacement divided by time elapsed:
v_avg = Ξx / Ξt
Average speed is total distance divided by time.
Like displacement, velocity is a vector. Speed is a scalar.
Instantaneous velocity is the velocity at one specific moment. Graphically, it equals the slope of the position-time graph at that moment.
Think of looking at your speedometer β that's your instantaneous speed. Your average speed on a road trip accounts for all the stopping and starting.
Position-Time Graph
Watch how position changes over time. The slope of the line equals the velocity.
βοΈ Worked Example
Problem: A car travels 120 km east in 2 hours, then 40 km west in 1 hour. Find (a) average speed, (b) average velocity for the entire trip.
β οΈ Common Mistakes
Misconception: Displacement and distance are the same thing.
β Correct thinking: Displacement is the straight-line change in position (a vector); distance is the total path length (a scalar). They are only equal when motion is in one direction with no backtracking.
Why: Confusing the two leads to wrong answers on any problem involving a return trip or a curved path.
Misconception: Average velocity equals (v_i + v_f) / 2 in all situations.
β Correct thinking: That formula only works when acceleration is constant. In general, average velocity = total displacement / total time.
Why: If the object speeds up and slows down unevenly, the simple average of endpoints does not equal the true average velocity.
Misconception: A negative velocity means the object is decelerating.
β Correct thinking: Negative velocity simply means the object is moving in the negative direction. Deceleration means the speed is decreasing, which happens when velocity and acceleration have opposite signs.
Why: Sign convention is tied to the coordinate system you chose, not to whether the object is speeding up or slowing down.
π Practice Problems
Try these problems. Check your answer when ready.
A jogger runs 600 m north, then turns around and runs 200 m south. What is the total distance traveled? What is the displacement?
A car travels 90 km in 1.5 hours. What is its average speed in m/s?
On a position-time graph, a straight line goes from (0 s, 2 m) to (4 s, 10 m). What is the object's velocity?
v = (Ξ x)/(Ξ t) = (10 - 2)/(4 - 0)A cyclist rides east at 12 m/s for 30 s, then west at 8 m/s for 15 s. Find (a) total distance, (b) net displacement, (c) average velocity for the whole trip.
A position-time graph shows a curve that gets steeper over time. What does this tell you about the velocity? About the acceleration?
An object's position is given by x(t) = 3tΒ² β 6t + 2 (meters, seconds). What is the object's average velocity between t = 1 s and t = 3 s?
v_avg = (x(3) - x(1))/(3 - 1)A train travels the first 100 km of a trip at 50 km/h and the next 100 km at 100 km/h. Is the average speed 75 km/h? Calculate the actual average speed.
Finished reading through this lesson?