Resistance kv on mass m: equation of motion?

m\frac{dv}{dt}=-kv, giving v=v0e^{-kt/m}

How do you find the constant of integration?

Substitute the initial conditions, such as v=u at t=0 or x=0.

Why can't suvat be used for a variable force?

The acceleration is not constant.

Sign of a resistance in the equation of motion?

Negative, because it opposes the motion.

Gravitational force at distance x from the Earth's centre?

\frac{mgR^2}{x^2}, with R the radius of the Earth

At the surface x=R the force is mg=\frac{GMm}{R^2}.

Variable forcesEdexcel International A Level Further Maths: Flashcards

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Three forms of acceleration?

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Three forms of acceleration?: $\frac{dv}{dt}$ , $\frac{d^2x}{dt^2}$ , $v\frac{dv}{dx}$
Which form for a force depending on displacement?: $a=v\frac{dv}{dx}$
Which form for a force depending on time?: $a=\frac{dv}{dt}$
Resistance $kv$ on mass $m$ : equation of motion?: $m\frac{dv}{dt}=-kv$ , giving $v=v_0e^{-kt/m}$
How do you find the constant of integration?: Substitute the initial conditions, such as $v=u$ at $t=0$ or $x=0$ .
Why can't suvat be used for a variable force?: The acceleration is not constant.
Sign of a resistance in the equation of motion?: Negative, because it opposes the motion.
Gravitational force at distance $x$ from the Earth's centre?: $\frac{mgR^2}{x^2}$ , with $R$ the radius of the Earth
Why is $GM=gR^2$ ?: At the surface $x=R$ the force is $mg=\frac{GMm}{R^2}$ .
$v^2$ for radial motion starting at the surface with speed $u$ ?: $u^2-2gR+\frac{2gR^2}{x}$
Escape speed from the Earth's surface?: $\sqrt{2gR}$
What is $v$ at the greatest distance of a particle moving in a line?: Zero

Exam questions on Variable forces

A particle $P$ of mass $2$ kg moves in a straight line. At time $t$ seconds the resultant force on $P$ is $(6t+4)$ N in the direction of motion. $P$ is at rest at the point $O$ when $t=0$ .
Find the distance travelled by $P$ in the first $2$ seconds.2 marks
A particle $P$ of mass $0.5$ kg moves along the positive $x$ -axis on a smooth horizontal surface, starting from the origin $O$ with speed $6$ m s $^{-1}$ . When $P$ is at displacement $x$ metres from $O$ , the only horizontal force on $P$ is a resistance of magnitude $2x$ N acting towards $O$ .
Find the speed of $P$ when it is $2$ m from $O$ .2 marks
A particle $P$ of mass $3$ kg moves in a straight line. At time $t$ seconds, $P$ has speed $v$ m s $^{-1}$ and the only horizontal force acting on $P$ is a resistance of magnitude $6v$ N. When $t=0$ the speed of $P$ is $8$ m s $^{-1}$ .
Show that $v=8e^{-2t}$ .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).