Say I have an object, initially at the origin travelling straight up the y-axis at a speed of v, I can say that y=vt, where t is time, (distance=speed*time)
Also, say I have another object which is initially (t=0) located at a point on the x-axis
This second object is travelling at a speed of 0.5v, and it is going in a direction such that its velocity vector is always directed towards the first object, so it is ALWAYS following the first object. This will ultimately make a curve which will have an equation that can be found.
Now it is safe to say that the second object will never catch the first, but if I were to find the closest this second object will get to the first, i.e, find the minimum distance between the 2 objects at any given moment in time, this is what I would do: (Please confirm that this will work)
Say that I have already found the equation of the path of the second object, say it is given by y= x^3 - 1/x^2 (which it most likely won't be, I'm just using it as an example)
Also assume I have found vt in terms of x
Now, the distance between any two points can be found using the distance formula, ((x2-x1)^2 +(y2-y1)^2)^1/2
So I would sub in:
x2=x (any point along the 2nd object's curve)
x1=0 (as the first object travels up the y-axis)
y2=y= x^3 - 1/x^2 (path of the the second object)
y1=vt (for reasons explained above)
Once I substitute these in, I would derive it and solve for when the derivative =0 (to find the minimum)
Once I find the value for x where the distance formula gives me a minimum value, I would sub it back into the original equation to find the minimum distance.
Hopefully that all makes sense! If I haven't explained it too well please let me know.
Please do tell me whether or not that would work! Or if there is an easier or better way of finding the minimum distance at a certain time
Thanks.
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