When average equals instantaneous

For constant acceleration, the average velocity equals the instantaneous velocity at the midpoint time.

When average equals instantaneous0 min · Free lecture

In this lesson

  • For constant acceleration, the average velocity equals the instantaneous velocity at the midpoint time.

The mean value theorem of kinematics: somewhere the average is achieved — at the midpoint for constant a.

Under constant acceleration, v at the mid-instant of an interval equals the interval's average velocity. This powers the famous 'two cars equal distance in equal time' shortcuts.

For variable acceleration, this fails — fall back to integrals.

v(tmid)=x2−x1t2−t1v(t_{mid}) = \frac{x_2 - x_1}{t_2 - t_1}

Midpoint theorem

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)
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When average equals instantaneous — FemtoLearn.
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For constant acceleration, the average velocity equals the instantaneous velocity at the midpoint time.
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