State the vector formula for displacement with constant acceleration.

\mathbf s=\mathbf ut+\frac12\mathbf at^2.

How do you find the position vector from the displacement?

\mathbf r=\mathbf r0+\mathbf s, adding the initial position.

Give \mathbf s in terms of \mathbf u, \mathbf v and t.

\mathbf s=\frac12(\mathbf u+\mathbf v)t.

What is speed in terms of velocity components?

\sqrt{vx^2+vy^2}, the magnitude of \mathbf v.

How is velocity found from position?

\mathbf v=\frac{d\mathbf r}{dt}, differentiating each component.

How is acceleration found from velocity?

\mathbf a=\frac{d\mathbf v}{dt}=\frac{d^2\mathbf r}{dt^2}.

How is velocity found from acceleration?

\mathbf v=\int\mathbf a\,dt, with a vector constant of integration.

How is position found from velocity?

\mathbf r=\int\mathbf v\,dt, using the initial position to find \mathbf c.

What do you need to find the constant of integration?

An initial condition, such as the velocity or position at t=0.

When is a particle at rest?

When both components of the velocity are zero at the same time.

When is a particle moving parallel to \mathbf i?

When the \mathbf j-component of the velocity is zero (and the \mathbf i-component is not).

What is the distance of a particle from O?

\lvert\mathbf r\rvert=\sqrt{x^2+y^2}.

Motion in two dimensions with vectorsAQA A-Level Maths: Flashcards

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State the constant acceleration formula for velocity in vector form.

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State the constant acceleration formula for velocity in vector form.: $\mathbf v=\mathbf u+\mathbf at$ .
State the vector formula for displacement with constant acceleration.: $\mathbf s=\mathbf ut+\frac12\mathbf at^2$ .
How do you find the position vector from the displacement?: $\mathbf r=\mathbf r_0+\mathbf s$ , adding the initial position.
Give $\mathbf s$ in terms of $\mathbf u$ , $\mathbf v$ and $t$ .: $\mathbf s=\frac12(\mathbf u+\mathbf v)t$ .
What is speed in terms of velocity components?: $\sqrt{v_x^2+v_y^2}$ , the magnitude of $\mathbf v$ .
How is velocity found from position?: $\mathbf v=\frac{d\mathbf r}{dt}$ , differentiating each component.
How is acceleration found from velocity?: $\mathbf a=\frac{d\mathbf v}{dt}=\frac{d^2\mathbf r}{dt^2}$ .
How is velocity found from acceleration?: $\mathbf v=\int\mathbf a\,dt$ , with a vector constant of integration.
How is position found from velocity?: $\mathbf r=\int\mathbf v\,dt$ , using the initial position to find $\mathbf c$ .
What do you need to find the constant of integration?: An initial condition, such as the velocity or position at $t=0$ .
When is a particle at rest?: When both components of the velocity are zero at the same time.
When is a particle moving parallel to $\mathbf i$ ?: When the $\mathbf j$ -component of the velocity is zero (and the $\mathbf i$ -component is not).
What is the distance of a particle from $O$ ?: $\lvert\mathbf r\rvert=\sqrt{x^2+y^2}$ .

Exam questions on Motion in two dimensions with vectors

A particle $P$ moves in a horizontal plane with constant acceleration $(2\mathbf i-\mathbf j)$ m s $^{-2}$ , where $\mathbf i$ and $\mathbf j$ are perpendicular unit vectors. At time $t=0$ the velocity of $P$ is $(\mathbf i+3\mathbf j)$ m s $^{-1}$ and its position vector relative to a fixed origin $O$ is $(3\mathbf i+4\mathbf j)$ m.
Find the speed of $P$ when $t=2$ .2 marks
A particle moves in a plane so that at time $t$ seconds its position vector relative to a fixed origin $O$ is $\mathbf r=\left[(t^3-3t)\mathbf i+(4t-t^2)\mathbf j\right]$ m, where $\mathbf i$ and $\mathbf j$ are perpendicular unit vectors.
Find the value of $t$ at which the particle is moving parallel to $\mathbf i$ .2 marks
A particle moves in a plane with acceleration $\mathbf a=(4t\,\mathbf i-2\mathbf j)$ m s $^{-2}$ at time $t$ seconds, where $\mathbf i$ and $\mathbf j$ are perpendicular unit vectors. When $t=0$ the velocity of the particle is $(2\mathbf i+5\mathbf j)$ m s $^{-1}$ and its position vector relative to a fixed origin $O$ is $(\mathbf i+3\mathbf j)$ m.
Find the velocity of the particle at time $t$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).