Constant speed
04-11-2016, 03:53 PM
Post: #10
 Tugdual Senior Member Posts: 756 Joined: Dec 2013
RE: Constant speed
(04-11-2016 02:01 PM)Martin Hepperle Wrote:  The function $$y=sin(x)$$ has the slope $${dy\over dx}=cos(x)$$ which yields
$dy=cos(x) \cdot dx$
The arc length of a segment with components $$dx$$ by $$dy$$ is given by
$ds = \sqrt{dx^2+dy^2}$
Substituting $$dy$$ into the above yields
$ds = \sqrt{dx^2+(cos(x) \cdot dx)^2}$
and solving for $$dx$$ gives
$dx = {ds \over \sqrt{cos(x)^2 + 1}}$
For any station $$x$$ this defines the required $$dx$$ to obtain the desired $$ds$$. Useful for plotting curves using constant segment lengths or constant speed driving on a curved road ;-)
The "General" answer depends on the equation - not all have simple Solutions for $$dx$$. Then some nonlinear solver resp. iteration would be needed.
Thanks Martin, sounds like a good answer but I struggle with further conclusions.
First, at constant speed, I would assume that ds/dt = 0 so dx = 0?
Other question is how do you move from your expressions to a parametric equation for x(t) and y(t) over the time?
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