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8.4MechanicsStretch

Differentiating Vectors

When a position vector is given as a function of time, you differentiate each component to get velocity, then again for acceleration. This handles variable acceleration in two dimensions.

24 min Further Kinematics
Edexcel A Level Maths: 8.4 Differentiating VectorsCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Differentiate a vector component-wise
  • Find velocity from a position vector
  • Find acceleration from velocity
  • Evaluate speed at a given time
1

Component-wise differentiation

If r(t)=x(t)i+y(t)j\mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j}, then v=x˙i+y˙j\mathbf{v} = \dot{x}\mathbf{i} + \dot{y}\mathbf{j} and a=x¨i+y¨j\mathbf{a} = \ddot{x}\mathbf{i} + \ddot{y}\mathbf{j} — differentiate each component separately.

v=drdt,a=dvdt\mathbf{v} = \frac{d\mathbf{r}}{dt}, \quad \mathbf{a} = \frac{d\mathbf{v}}{dt}
Differentiate each component.
Example 1If r=t2i+(3t)j\mathbf{r} = t^2\mathbf{i} + (3t)\mathbf{j}, find v\mathbf{v} and a\mathbf{a}.
1v=2ti+3j\mathbf{v} = 2t\mathbf{i} + 3\mathbf{j}.
2a=2i\mathbf{a} = 2\mathbf{i}.
Answer
v=2ti+3j\mathbf{v} = 2t\mathbf{i} + 3\mathbf{j}, a=2i\mathbf{a} = 2\mathbf{i}.

Tip — Speed is the magnitude of the velocity vector: |v| = √(vₓ² + v_y²).

Formula recap

v=x˙i+y˙j\mathbf{v} = \dot{x}\mathbf{i} + \dot{y}\mathbf{j}
Velocity.
a=x¨i+y¨j\mathbf{a} = \ddot{x}\mathbf{i} + \ddot{y}\mathbf{j}
Acceleration.

Common mistakes to avoid

Differentiating the magnitude instead of each component.
Differentiate the i and j components separately.
Confusing speed with velocity.
Speed is the magnitude of the velocity vector.

Key takeaways

  • Differentiate r component-wise to get v, then a.
  • v = ẋ i + ẏ j; a = ẍ i + ÿ j.
  • Speed = |v|.

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 a position vector you:

    • differentiate each component
    • integrate
    • take the magnitude
    • add components
    Show answer and explanation

    Answer: differentiate each component

    v = dr/dt.

  2. If r=t2i+3tj\mathbf{r} = t^2\mathbf{i} + 3t\mathbf{j}, then v=\mathbf{v} =

    • 2ti+3j2t\mathbf{i} + 3\mathbf{j}
    • t2i+3jt^2\mathbf{i} + 3\mathbf{j}
    • 2i2\mathbf{i}
    • 2ti2t\mathbf{i}
    Show answer and explanation

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

    Differentiate.

  3. For that r, the acceleration is:

    • 2i2\mathbf{i}
    • 2ti2t\mathbf{i}
    • 3j3\mathbf{j}
    • 00
    Show answer and explanation

    Answer: 2i2\mathbf{i}

    d/dt of v.

  4. Speed is:

    • the magnitude of velocity
    • the velocity vector
    • the acceleration
    • the displacement
    Show answer and explanation

    Answer: the magnitude of velocity

    |v|.

  5. Acceleration is found by differentiating:

    • the velocity vector
    • the position once
    • the speed
    • nothing
    Show answer and explanation

    Answer: the velocity vector

    a = dv/dt.

Next step

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