3.5 Linear motion under a variable force

Learning objectives

3.5.1Variable force motion• solve problems which can be modelled as the linear motion of a particle under the action of a variable force, by setting up and solving an appropriate differential equation. Including use of v x v d d for acceleration, where appropriate. Calculus required is restricted to content from Pure Mathematics 3 in Cambridge International A Level Mathematics (9709). Only differential equations in which the variables are separable are included.

force/result variables	acceleration form in $m a=\sum F$	usual next step
time $t$ and velocity $v$	$a=dv/dt$	separate to find $v(t)$ , then use $dx/dt=v$
position $x$ and velocity $v$	$a=v\,dv/dx$	separate to find $v(x)$
position only and speed required	$a=v\,dv/dx$	integrate directly with the position conditions

Choose a positive direction and give every real force its signed component. Write Newton's second law before cancelling mass. Separate variables, integrate within Pure Mathematics 3 methods, and use the condition at a known time or position to determine the constant. Only separable differential equations are required.

$A particle of mass $m$ moves positively with $v(0)=1$ and experiences resistance $mkv^3$. Then$ m\frac{dv}{dt}=-mkv^3,\qquad v^{-3}dv=-k,dt. $Hence$ -\frac{1}{2v^2}=-kt+C. $Using $v=1$ at $t=0$ gives $C=-\tfrac12$, so$ v(t)=\frac{1}{\sqrt{1+2kt}}. $Displacement follows from integrating $dx/dt=v(t)$ with its own position condition.$

Do not use constant-acceleration formulae when the resultant varies. In v dv/dx, v is signed velocity, so state the motion interval before choosing a root. A stopping point may be approached asymptotically; check the solved expression or definite integral rather than assuming a finite time.

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