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8.5MechanicsStretch

Integrating Vectors

Integration reverses differentiation for vectors too. From an acceleration vector you integrate to velocity, and again to position — using initial conditions to fix the vector constants.

24 min Further Kinematics
Edexcel A Level Maths: 8.5 Integrating VectorsCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Integrate a vector component-wise
  • Find velocity from acceleration
  • Find position from velocity
  • Use initial conditions for the constant vector
1

Component-wise integration

Integrate each component and add a constant vector\textbf{vector}. v=∫a dt+c\mathbf{v} = \int \mathbf{a}\,dt + \mathbf{c}, with c\mathbf{c} found from the velocity at a known time.

v=∫a dt+c\mathbf{v} = \int \mathbf{a}\,dt + \mathbf{c}
Integrate each component; add a constant vector.
Example 1If a=2i\mathbf{a} = 2\mathbf{i} and v=3j\mathbf{v} = 3\mathbf{j} at t=0t = 0, find v(t)\mathbf{v}(t).
1∫2i dt=2ti+c\int 2\mathbf{i}\,dt = 2t\mathbf{i} + \mathbf{c}.
2At t=0t=0, v=3j\mathbf{v} = 3\mathbf{j}, so c=3j\mathbf{c} = 3\mathbf{j}.
Answer
v=2ti+3j\mathbf{v} = 2t\mathbf{i} + 3\mathbf{j}

Tip — The constant of integration is a VECTOR — find it from the initial velocity or position.

Formula recap

v=∫a dt+c\mathbf{v} = \int \mathbf{a}\,dt + \mathbf{c}
Velocity from acceleration.
r=∫v dt+c\mathbf{r} = \int \mathbf{v}\,dt + \mathbf{c}
Position from velocity.

Common mistakes to avoid

Adding a scalar constant of integration.
The constant is a vector (it has i and j parts).
Skipping the initial conditions.
Use the given velocity/position to find the constant vector.

Key takeaways

  • Integrate each component, add a constant vector.
  • v = ∫a dt + c; r = ∫v dt + c.
  • Use initial conditions to find the constant vector.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. To find velocity from acceleration you:

    • integrate
    • differentiate
    • take the magnitude
    • use suvat only
    Show answer and explanation

    Answer: integrate

    v = ∫a dt.

  2. The constant of integration here is:

    • a vector
    • a scalar
    • always zero
    • the speed
    Show answer and explanation

    Answer: a vector

    Has i and j parts.

  3. With a=2i\mathbf{a}=2\mathbf{i}, v=3j\mathbf{v}=3\mathbf{j} at t=0, v(t)=\mathbf{v}(t)=

    • 2ti+3j2t\mathbf{i}+3\mathbf{j}
    • 2ti2t\mathbf{i}
    • 3j3\mathbf{j}
    • 2i+3j2\mathbf{i}+3\mathbf{j}
    Show answer and explanation

    Answer: 2ti+3j2t\mathbf{i}+3\mathbf{j}

    c = 3j.

  4. Position is found by integrating:

    • the velocity
    • the acceleration once
    • the speed
    • the displacement
    Show answer and explanation

    Answer: the velocity

    r = ∫v dt.

  5. The constant vector is determined using:

    • initial conditions
    • the magnitude
    • the average
    • nothing
    Show answer and explanation

    Answer: initial conditions

    Given v or r at a time.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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