Motion in a Straight Line · Lesson 3 of 10
Instantaneous Velocity and Speed
“Move from interval averages to tangents, limits and derivatives, with numerical and algebraic examples.”
• Explain instantaneous velocity as the limiting average velocity. • Read velocity from a tangent to a position-time curve. • Differentiate simple position functions. • Distinguish instantaneous speed from average speed.
What a speedometer tells you
A journey’s average speed can hide large variations. A speedometer aims to show how fast the vehicle is moving now. To make this precise, calculate average velocity over shorter and shorter intervals around the chosen instant. If those values approach one value, that limit is the instantaneous velocity. We usually shorten the name to velocity when no interval average is intended.
The notation dx/dt means the derivative of position with respect to time, the local rate of change. Graphically, a secant through nearby points becomes a tangent at the selected point. Calculate the tangent’s rise divided by its run using two convenient points on that tangent. They need not both lie on the original curved graph. Also account for the numerical scales on both axes; a visually steeper line is not enough if scales differ.
Watching the limit emerge numerically
Consider x = 0.08t³, with x in metres and t in seconds. To estimate the velocity at 4 s, centre an interval of width Δt on 4 s. Its endpoints are 4 − Δt/2 and 4 + Δt/2. Substituting both times gives an average velocity approaching a definite limit.
| Interval Δt (s) | Initial time (s) | Final time (s) | Average velocity (m/s) |
|---|---|---|---|
| 2 | 3 | 5 | 3.92 |
| 1 | 3.5 | 4.5 | 3.86 |
| 0.5 | 3.75 | 4.25 | 3.845 |
| 0.1 | 3.95 | 4.05 | 3.8402 |
| 0.01 | 3.995 | 4.005 | 3.840002 |
The values approach 3.84 m/s. Algebra explains why: the centred quotient is 0.24t² + 0.02(Δt)². At t = 4 s, the first term is 3.84 and the second term tends to zero. Finite intervals produce an approximation; the limit defines the exact instantaneous velocity of this model.
Differentiation as a useful shortcut
For a constant C, dC/dt = 0 because it does not change. For Ctⁿ, the derivative is nCtⁿ⁻¹. Apply the rule to each term and keep the units consistent. If x = 8.5 + 2.5t² is measured in metres, 8.5 carries units m and the coefficient 2.5 carries units m/s². The resulting velocity has units m/s, as it must.
Problem
For x = 8.5 + 2.5t², find velocity at 0 s and 2 s, then average velocity from 2 s to 4 s.
- 1.Differentiating gives v = 5t, with v in m/s when t is in s.
- 2.At 0 s, v = 0. At 2 s, v = 10 m/s.
- 3.The endpoint positions are x(2) = 18.5 m and x(4) = 48.5 m. Average velocity is (48.5 − 18.5)/(4 − 2) = 15 m/s.
- 4.This interval average equals the instantaneous velocity at 3 s here because velocity is linear in time. It is not a general rule for arbitrary motion.
Velocity has a direction; speed does not
Over a sufficiently short smooth portion of one-dimensional motion without a reversal, distance is the magnitude of displacement. Taking the limiting rate therefore gives speed = |velocity|. At a smooth turning point both instantaneous velocity and speed are zero, even though both can have nonzero averages over a surrounding interval. This is why the distinction between average speed and |average velocity| does not carry over to instantaneous values.
Problem
An object has x = 6t − t² metres. Describe its motion near t = 3 s.
- 1.Velocity is v = 6 − 2t m/s. It is positive before 3 s, zero at 3 s and negative after 3 s.
- 2.The position at 3 s is 9 m, a maximum. The object reaches that position and reverses direction.
- 3.At 4 s, velocity is −2 m/s and speed is 2 m/s. Negative velocity says which way it moves, not that it moves slowly.
- 4.A horizontal tangent only proves instantaneous rest. To prove reversal, check that velocity changes sign across the instant.
Uniform motion and idealised corners
With constant velocity, x = x₀ + vt is a straight line. Every secant and tangent has the same slope, so average and instantaneous velocity are identical for every interval. A flat line means rest. At a sharp corner in an idealised position-time graph, the left and right slopes differ, so a unique derivative does not exist exactly there. Real changes occur over finite times; such corners are simplifying approximations, not places to invent a single velocity.
Quiz
For x = 2 + 3t² metres, what is the instantaneous velocity at t = 2 s?
Key Takeaways
• Instantaneous velocity is a limit of interval averages and the slope of a tangent. • Differentiate position to obtain velocity. • Speed is the magnitude of instantaneous velocity. • A horizontal tangent means instantaneous rest; a sign change establishes reversal.