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suppose i have x[q], y[q]=S, the surface of a hill

dy/dx=(dy/dt)/(dx/dt)

atan(dy/dx)= the angle of incline at a location on the hill

-GM*cos(pi/2-atan(dy/dx))=the tangeantal component of acceleration

im ignoring the perpendicular component because theres no friction here.

-GM*sin(atan(dy/dx)=d2(x^2+y^2)^.5/dt2

i guess i could apply conservation of energy and get

E=.5(dy/dt)^2+.5(dx/dt)^2+gm*INT(sin(atan(dy/dx)))d(x^2+y^2)^.5

but dy/dx isnt neccessarily a function of x^2+y^2, so im kind of out of luck

and even then i dont have a second conserved quantity(such as angular momentum like in the central force problem), which would be useful.

i resolved the components again and got this system

-gm*cos(atan(dy/dx))*sin(atan(dy/dx))=d2x/dt2

-gm*(sin(atan(dy/dx))^2=d2y/dt2

So, how do i solve this problem where the slope of the hill is not a constant?

I want to find how fast an object will be, the time taken, etc, for an object to go from one position to another on the hill. how do i do this?