What is acceleration in terms of velocity and displacement?

a=\dfrac{dv}{dt}=\dfrac{d^2x}{dt^2}

How do you find velocity from acceleration a(t)?

Integrate: v=\int a\,dt, then use an initial condition for the constant.

How do you find the time of greatest velocity?

Solve a=0, then check the end points.

Can you use suvat when a depends on t?

No; suvat needs constant acceleration.

What is the difference between speed and velocity?

Speed is the magnitude of velocity and is never negative.

How do you find the total distance when the particle changes direction?

Split at each time v=0 and add the magnitudes of the separate displacements.

What does \int{t1}^{t2}v\,dt give?

The displacement between t1 and t2.

Velocity from a position vector \mathbf{r}?

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

\mathbf{r}=t^3\mathbf{i}+2t^2\mathbf{j}: find \mathbf{v} and \mathbf{a}.

\mathbf{v}=3t^2\mathbf{i}+4t\mathbf{j}, \mathbf{a}=6t\mathbf{i}+4\mathbf{j}

Why must you include a constant when integrating?

Initial conditions fix the constant; omitting it gives the wrong velocity or displacement.

Kinematics using calculusEdexcel International A Level Further Maths: Flashcards

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What is velocity in terms of displacement?

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What is velocity in terms of displacement?: $v=\dfrac{dx}{dt}$
What is acceleration in terms of velocity and displacement?: $a=\dfrac{dv}{dt}=\dfrac{d^2x}{dt^2}$
How do you find velocity from acceleration $a(t)$ ?: Integrate: $v=\int a\,dt$ , then use an initial condition for the constant.
How do you find when a particle is at rest?: Solve $v=0$ .
How do you find the time of greatest velocity?: Solve $a=0$ , then check the end points.
Can you use suvat when $a$ depends on $t$ ?: No; suvat needs constant acceleration.
What is the difference between speed and velocity?: Speed is the magnitude of velocity and is never negative.
How do you find the total distance when the particle changes direction?: Split at each time $v=0$ and add the magnitudes of the separate displacements.
What does $\int_{t_1}^{t_2}v\,dt$ give?: The displacement between $t_1$ and $t_2$ .
Velocity from a position vector $\mathbf{r}$ ?: $\mathbf{v}=\dfrac{d\mathbf{r}}{dt}$ , differentiating each component.
How do you find the speed from a velocity vector?: $|\mathbf{v}|=\sqrt{v_x^2+v_y^2}$
$\mathbf{r}=t^3\mathbf{i}+2t^2\mathbf{j}$ : find $\mathbf{v}$ and $\mathbf{a}$ .: $\mathbf{v}=3t^2\mathbf{i}+4t\mathbf{j}$ , $\mathbf{a}=6t\mathbf{i}+4\mathbf{j}$
Why must you include a constant when integrating?: Initial conditions fix the constant; omitting it gives the wrong velocity or displacement.

Exam questions on Kinematics using calculus

A particle $P$ moves along a straight line. Its displacement $x$ metres from a fixed point $O$ at time $t$ seconds is given by $x=t^3-6t^2+9t$ , for $t\geq0$ .
Find the displacement of $P$ from $O$ at the instant when its acceleration is zero.2 marks
A particle $Q$ moves along a straight line through a fixed point $O$ . At time $t$ seconds its velocity is $v=6t-3t^2$ in $\text{m s}^{-1}$ , for $0\leq t\leq3$ , and $Q$ is at $O$ when $t=0$ .
Find the greatest velocity of $Q$ for $0\leq t\leq3$ .2 marks
A particle $P$ moves in a horizontal plane. At time $t$ seconds its position vector relative to a fixed origin $O$ is $\mathbf{r}=(2t^3-3t)\mathbf{i}+(t^2+4t)\mathbf{j}$ metres, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors.
Find the velocity of $P$ when $t=2$ .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).