M10.4MechanicsStretch

Motion in 2 Dimensions

Newton’s second law works as a vector equation too: F = ma in i–j form. This lets you find acceleration from forces given as vectors, and combine it with vector SUVAT for full 2D motion.

30 min Forces and Motion

Edexcel AS Level Maths: 10.4 Motion in 2 DimensionsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Apply F = ma in vector (i–j) form
Find acceleration as a vector
Find the magnitude of acceleration
Link vector force to vector motion

F = ma as vectors

When the resultant force is a vector, the acceleration is a vector in the same direction: $\mathbf{F} = m\mathbf{a}$ . Divide the force vector by the mass to get the acceleration vector.

\mathbf{F} = m\mathbf{a} \;\Rightarrow\; \mathbf{a} = \dfrac{1}{m}\mathbf{F}

Same direction; divide each component by

m

Example 1A resultant force

(6\mathbf{i} + 8\mathbf{j})

N acts on a 2 kg mass. Find the acceleration.

\mathbf{a} = \tfrac{1}{2}(6\mathbf{i} + 8\mathbf{j})

Answer

3\mathbf{i} + 4\mathbf{j}

m/s²

Magnitude of acceleration

The size of the acceleration is the magnitude of the acceleration vector, found with Pythagoras.

|\mathbf{a}| = \sqrt{a_x^2 + a_y^2}

e.g.

|3\mathbf{i} + 4\mathbf{j}| = 5

m/s².

Combining with vector SUVAT

Once you have the acceleration vector, the vector forms of SUVAT (e.g. $\mathbf{v} = \mathbf{u} + \mathbf{a}t$ ) describe the full 2D motion — velocity and position as vectors over time.

Tip — Resolve forces and motion into i and j, then treat each direction with the usual rules.

Formula recap

\mathbf{F} = m\mathbf{a}

Vector form of Newton’s 2nd law.

|\mathbf{a}| = \sqrt{a_x^2 + a_y^2}

Magnitude of acceleration.

\mathbf{v} = \mathbf{u} + \mathbf{a}t

Vector SUVAT.

Common mistakes to avoid

Dividing only one component of the force by the mass.

Divide every component by m to get the acceleration vector.

Adding i and j components when finding magnitude.

Magnitude uses √(aₓ² + a_y²), not aₓ + a_y.

Key takeaways

F = ma holds as a vector equation: a = (1/m)F.
Acceleration magnitude = √(aₓ² + a_y²).
Combine with vector SUVAT (v = u + at) for full 2D motion.

Test yourself

Answer a few questions, then review anything you missed.

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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