Q3MechanicsStretch

Vectors in Kinematics

Motion in two dimensions is described with vectors. Position, velocity and acceleration become vector quantities, and the constant-acceleration (suvat) equations work in vector form.

30 min Kinematics

The big picture

For constant vector acceleration, $\mathbf r=\mathbf r_0+\mathbf ut+\tfrac12\mathbf at^2$ . A collision requires equality of both coordinates at the same time, not just an intersection of the two geometric paths. For a moving particle to be due east or west of a point, set the vertical separation to zero, then check the sign of the horizontal separation. Restrict time to the interval where the model applies.

Edexcel A Level Maths: 8.1 Vectors in Kinematics (Part 1)Curated by Kepler, not produced by us. Watch it, then work through the notes below.

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What you'll be able to do

Use position, velocity and acceleration vectors
Apply suvat in vector form
Find displacement and velocity at a time
Work with i, j components

Vector suvat

For constant acceleration, the suvat equations hold as vectors. For example $\mathbf{v} = \mathbf{u} + \mathbf{a}t$ and $\mathbf{s} = \mathbf{u}t + \tfrac12\mathbf{a}t^2$ , each applied component by component.

\mathbf{v} = \mathbf{u} + \mathbf{a}t, \quad \mathbf{s} = \mathbf{u}t + \tfrac12\mathbf{a}t^2

suvat in vector form.

Example 1A particle has

\mathbf{u} = 3\mathbf{i} + \mathbf{j}

and

\mathbf{a} = \mathbf{i} - 2\mathbf{j}

. Find

\mathbf{v}

t = 2

\mathbf{v} = (3\mathbf{i}+\mathbf{j}) + 2(\mathbf{i}-2\mathbf{j})

= 5\mathbf{i} - 3\mathbf{j}

Answer

5\mathbf{i} - 3\mathbf{j}

Tip — Apply each suvat equation separately to the i and j components.

Simultaneous positions and a physical time

For a moving particle to be due east or west of a point, set the vertical separation to zero, then check the sign of the horizontal separation. Restrict time to the interval where the model applies.

Example 1Particles have positions

\mathbf r_A=(1+2t)\mathbf i+3t\mathbf j

and

\mathbf r_B=(7-t)\mathbf i+(2+t)\mathbf j

. Do they collide for t >= 0?

1Equal x-coordinates give

1+2t=7-t

, so t = 2.

2Equal y-coordinates give

3t=2+t

, so t = 1.

3These are different times, so both coordinates are never equal simultaneously.

AnswerThey do not collide. Coordinate agreement must occur at one common time.

Think like an examiner

Write the defining equations and show the substitution before evaluating.
Check the domain, units and interpretation of the final result.

Common misconceptions

Adding the magnitudes of vectors directly.

Work component by component.

Forgetting suvat only applies for constant acceleration.

Use calculus methods when acceleration varies.

Key results to remember

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

Velocity.

\mathbf{s} = \mathbf{u}t + \tfrac12\mathbf{a}t^2

Displacement.

Key takeaways

Position, velocity, acceleration are vectors.
suvat holds in vector form for constant acceleration.
Work component by component (i and j).

Test yourself

Answer a few questions, then review anything you missed.

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