How do you get v from a(t)?

Integrate: v=\int a\,dt, and use the initial velocity to find the constant.

Why can't suvat be used with variable acceleration?

The suvat equations assume constant acceleration.

How to solve \frac{dv}{dt}=f(v)?

Separate the variables: \int\frac{dv}{f(v)}=\int dt.

What is limiting speed?

The speed approached as t\to\infty, where the acceleration is zero.

Difference between distance and displacement?

Displacement is the change in position (with sign); distance is total path length, so split the motion where v=0.

Variable acceleration in one dimensionEdexcel A-Level Further Maths: Flashcards

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Relationships between $x$, $v$ and $a$ using calculus?

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Relationships between $x$ , $v$ and $a$ using calculus?: $v=\frac{dx}{dt}$ and $a=\frac{dv}{dt}$ .
How do you get $v$ from $a(t)$ ?: Integrate: $v=\int a\,dt$ , and use the initial velocity to find the constant.
Why can't suvat be used with variable acceleration?: The suvat equations assume constant acceleration.
How do you find when a particle is instantaneously at rest?: Solve $v=0$ .
How to solve $\frac{dv}{dt}=f(v)$ ?: Separate the variables: $\int\frac{dv}{f(v)}=\int dt$ .
$\int e^{kt}\,dt$ ?: $\frac1ke^{kt}+c$ .
$\int\cos kt\,dt$ ?: $\frac1k\sin kt+c$ .
What angle mode must a calculator be in for $\sin2t$ ?: Radians.
What is limiting speed?: The speed approached as $t\to\infty$ , where the acceleration is zero.
Solution of $\frac{dv}{dt}=-kv$ with $v=u$ at $t=0$ ?: $v=ue^{-kt}$ .
Difference between distance and displacement?: Displacement is the change in position (with sign); distance is total path length, so split the motion where $v=0$ .
How do you write a deceleration of $0.02v^2$ ?: $\frac{dv}{dt}=-0.02v^2$ .

Exam questions on Variable acceleration in one dimension

A particle $P$ moves in a straight line. At time $t$ seconds, where $t\ge0$ , its acceleration is $(6t-12)$ m s⁻² in the positive direction. At $t=0$ , $P$ is at the origin $O$ with velocity 9 m s⁻¹ in the positive direction.
Find the times at which $P$ is instantaneously at rest.2 marks
A sports car accelerates from rest along a straight, level track. At time $t$ seconds its velocity is $v$ m s⁻¹ and its acceleration is $\frac{60-v}{20}$ m s⁻².
Find the time taken for the car to reach 90% of its limiting speed.2 marks
A particle $P$ moves along the $x$ -axis. At time $t$ seconds its velocity in the positive $x$ -direction is $6\cos2t$ m s⁻¹. At $t=0$ , $P$ is at the point with coordinate $x=1$ . Angles are in radians.
Find an expression for $x$ in terms of $t$ .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).