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If a 1kg object under constant acceleration, g, starts out at rest and over the course of .5 seconds traverses 1m winding up at rest again, can you deduce exactly force F that acted on it? Or only the impulse?

Given:

Δp = 0

Δke = 0

Δx = 1

Δt = .5

g = 9.8

unknowns:

F - the unknown force

t

_{1}- time F was applied

x

_{1}- distance F was applied

Equations:

gt = Ft

_{1}

gx = Fx

_{1}

That gives me x/t = x

_{1}/t

_{1}

and x

_{1}= .5((F-g)/m)t

_{1}

^{2}

I am having trouble working it out, but it seems at first glance that since x

_{1}= .5((F-g)/m)t

_{1}

^{2}is not linear there would be only 1 set of appropriate F, x

_{1}and t

_{1}that get me exactly 1m in .5s and back at rest. Any ideas?