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 Video by Zeeshan Zamurred Further Kinematics

Edexcel A Level Maths: 8.4 Differentiating VectorsWatch the full walkthrough 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

Component-wise differentiation

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

v = \frac{d r}{d t}, a = \frac{d v}{d t}

Differentiate each component.

v = 2 t i + 3 j

a = 2 i

Answer

v = 2 t i + 3 j

a = 2 i

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

Formula recap

v = \overset{x}{˙} i + \overset{y}{˙} j

Velocity.

a = \overset{x}{¨} i + \overset{y}{¨} 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|.

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