Instantaneous velocity and speed
Position, path length and displacement
A car drives along a straight road. To describe where it is, we first need a reference point. Let us call it O, the origin.
Draw an x-axis along the road through O. Points to the right of O have positive x. Points to the left have negative x.
The car starts at O and drives right to point P, at x = +360 m. Then it turns back and stops at Q, at x = +240 m.
How far did it actually drive? 360 m out, then 120 m back. The path length is 480 m. Path length is always positive.
Now just compare the start and the end. The car began at 0 and finished at +240 m. That change in position, Δx = x₂ − x₁, is the displacement.
Watch the single arrow from O to Q. Displacement has a size and a direction, and the sign of Δx tells the direction. Here it is +240 m.
If the car came all the way back to O, the path length would be 720 m, but the displacement would be zero.
From average to instantaneous velocity
Say the trip from O to P takes 18 s, and the drive back to Q takes 6 s more. The whole trip takes 24 s.
Average velocity is displacement divided by time: 240 m over 24 s, which is +10 m/s. Average speed is path length divided by time: 480 m over 24 s, which is 20 m/s.
So average speed and the size of average velocity are not always equal. They match only when the car never turns back.
Now plot the car's position x against time t. The x–t graph of a car that is speeding up is a curve that gets steeper.
Pick two points on the curve and join them with a straight line, called a chord. The slope of the chord, Δx over Δt, is the average velocity over that interval.
Now slide the second point closer and closer to the first. Watch the chord swing round until it just touches the curve. It becomes the tangent.
The slope of the tangent is the instantaneous velocity: v = dx/dt, the limit of Δx/Δt as Δt goes to zero. A speedometer shows its size, the instantaneous speed.
Try it: slope of the x–t graph
- Press Play, or drag the Time slider to scrub through the motion.
- The orange line is the tangent: its slope is the instantaneous velocity v.
- The dashed blue chord joins t = 0 to now: its slope is the average velocity.
- Set a = 0 and the x–t graph becomes a straight line. Now the tangent and chord have the same slope.
- Make u negative and see the car drive left, with a downward-sloping x–t graph.
Quick check
A car goes 360 m east in 18 s, then turns around and goes 120 m west in 6 s. What are its average speed and average velocity for the whole trip?
- 20 m/s, and 10 m/s east
- 10 m/s, and 10 m/s east
- 20 m/s, and 20 m/s east
- 20 m/s, and 15 m/s east
Answer
A. Path length = 360 + 120 = 480 m, so average speed = 480 m / 24 s = 20 m/s. Displacement = 360 − 120 = 240 m east, so average velocity = 240 m / 24 s = 10 m/s east. The two differ because the car turned back.
Reading an x–t graph
The x–t graph of a car is a curve that gets steeper and steeper as time goes on. What is happening to the car?
- It is moving at a constant velocity
- Its velocity is increasing
- Its velocity is decreasing
- It is climbing up a hill
Answer
B. The slope of the x–t graph at any instant is the instantaneous velocity. A curve that gets steeper has a slope that keeps growing, so the velocity is increasing. An x–t graph is not a picture of the road, so it says nothing about hills.
Summary
- Displacement:
Δx = x₂ − x₁ - Average velocity:
v̄ = Δx / Δt = (x₂ − x₁) / (t₂ − t₁) - Average speed:
average speed = total path length / total time - Instantaneous velocity:
v = lim (Δt → 0) Δx/Δt = dx/dt
- Path length is the total distance driven. Displacement is just the change in position, and it can be positive, negative or zero.
- Average velocity is the slope of the chord on the x–t graph. Instantaneous velocity is the slope of the tangent.
- Instantaneous speed is the size of the instantaneous velocity, |v|. Over a trip, average speed can be more than the size of the average velocity, but never less.