Instantaneous vs average quantities

Instantaneous = at one instant (derivative); average = over an interval (quotient). Never mix them.

Instantaneous vs average quantities0 min · Free lecture

In this lesson

  • Instantaneous = at one instant (derivative); average = over an interval (quotient). Never mix them.

Your speedometer (instantaneous) is rarely your trip average — and both are 'the speed'.

v_inst = dx/dt at a point; v_avg = Δx/Δt over an interval. Similarly a_inst = dv/dt, a_avg = Δv/Δt.

Questions asking 'at t = 2 s' want instantaneous values; questions about 'between t = 2 and t = 4' want averages.

v=lim⁡Δt→0ΔxΔtv = \lim_{\Delta t\to 0}\frac{\Delta x}{\Delta t}

Instantaneous velocity

Worked example

The acceleration of a body starting from rest is a = 2 m/s² for 4 s, then zero for 2 s. Find its velocity at t = 6 s.

  1. Area under the a–t graph = change in velocity.
  2. Area = 2 m/s² × 4 s = 8 m/s (the last 2 s add nothing).

Answer: 8

Transcript (0 min)
WEBVTT

1
00:00:00.000 --> 00:00:05.000
Instantaneous vs average quantities — FemtoLearn.
2
00:00:05.000 --> 00:00:15.000
Instantaneous = at one instant (derivative); average = over an interval (quotient). Never mix them.
Printable notes (free)

Practice

Free · 4 questions with full solutions
  1. Q1 · Numerical · difficulty 2/5

    The acceleration of a body starting from rest is a = 2 m/s² for 4 s, then zero for 2 s. Find its velocity at t = 6 s.

  2. Q2 · MCQ · difficulty 2/5

    A particle moves with constant velocity. Which a–t graph describes it?

    • A horizontal line at a = 0
    • A horizontal line at a = 2 m/s²
    • A straight line through the origin
    • A parabola
  3. Q3 · Numerical · difficulty 2/5

    A body moves with v = 10 m/s for 3 s, then v = 5 m/s for 2 s in the same direction. Find the total distance.

  4. Q4 · MCQ · difficulty 2/5

    The slope of a velocity–time graph gives:

    • Displacement
    • Acceleration
    • Speed
    • Distance