= Balanced forces and constant velocity
{title2=$6\,\mathrm N-6\,\mathrm N=0$}
Consider a stipulated ideal cart modeled as a <point mass> of <mass> $2\,\mathrm{kg}$ on a horizontal track in a laboratory <inertial frame>. Its initial <position> is $x(0)=0$ and its initial <velocity> is $v_x(0)=1.5\,\mathrm{m/s}$ to the right. Independent devices exert horizontal <forces> $+6\,\mathrm N$ and $-6\,\mathrm N$ on this same cart. Its upward support <force> balances its downward <gravitational weight>; no other <forces> contribute. Rotation and deformation are excluded from this model.
$$
F_{\mathrm{net},x}=6-6=0\,\mathrm N,\qquad
a_x=\frac{0\,\mathrm N}{2\,\mathrm{kg}}=0\,\mathrm{m/s^2},\qquad
v_x(t)=1.5\,\mathrm{m/s},\qquad
x(t)=(1.5\,\mathrm{m/s})t.
$$
After $4\,\mathrm s$, the cart is at $6\,\mathrm m$, still moving right at $1.5\,\mathrm{m/s}$. Starting the same model at rest instead would leave it at rest. <Force balance> determines the zero <acceleration>, while the initial <velocity> determines whether the cart moves.
\Image[/inertial-cart-force-balance.png]
{title=Balanced <external forces> on one cart and its description in different <reference frames>}
{description=Original schematic of the stipulated $2\,\mathrm{kg}$ cart. The upper panel shows actual <forces> and a separate <velocity> arrow. The lower panels compare an <inertial frame> translating at constant <velocity> with a nonrotating <translating accelerated reference frame>.}
For the plotted model, both horizontal devices are switched off simultaneously at $t=2\,\mathrm s$, with no <impulse>. The support and <gravitational weight> remain balanced. The individual horizontal <forces> change, but their sum stays zero on both sides of the switch, so the <velocity> stays continuous and unchanged. Switching off just one device would be a different model with a nonzero net <external force>.
\Image[/inertial-cart-position-velocity-forces.png]
{title=Model <position>, <velocity>, and individual horizontal <forces> before and after a simultaneous switch}
{description=Exact predictions for the stipulated cart, not experimental data. For $0\le t<2\,\mathrm s$, the horizontal <forces> are $+6\,\mathrm N$ and $-6\,\mathrm N$; for $2\le t\le4\,\mathrm s$, both are zero. The net horizontal <external force> is zero throughout. Open and filled endpoints distinguish the two sides of the switch.}
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