So I was attempting to learn how to model trajectories and I cannot connect the dots on how to change the velocity of a trajectory following a path described by a function of time, since the velocity of that trajectory is then just the derivative of that position function with respect to time? for example an object follows the path described by the vector function $$ P(t) = (x(t), y(t), z(t)) = ( -250cos(t), 250sin(t), 100(t)) $$ This equation describes an object moving in a helical path upward at 100 ft per second and orbiting the origin with a radius of 250 ft. Now here is where I am stuck, the speed of this object is based on the previous equation and is $$ \mid V(t) \mid = P(t) dt = \sqrt{x(t)^2 + y(t)^2 + z(t)^2} = 269ft/s $$
and the radius it orbits the origin and the rate it rises is 250 and 100ft/s, how can I separate the path from the velocity so that I can describe objects that travel the same path but at 250ft/s or any other arbitrary speed? My thought was to somehow remove time from the equation and just end up with coordinates but then how would you add the new velocity you want back in? any thoughts or pointers would be appreciated.
Update:
So per @joe's comment I divided t/2 and that did change the velocity. So essentially that 250 is your x y and 100 the z so the distance travelled multiplied by the time would give your rate of change. So by dividing t/2 you end up multiplying the x y or z value by a fraction of the time t and travelling less over the same time period. ie slower speed. Now in order to know the specific divider value you need to convert the meters/sec or ft/sec to radians/sec which will give you a fraction that gets inserted into the trig functions this is how you relate the speed to the path equation.
With an original $ V(t) = ( 0 , 0 , 0 )$ and an initial velocity vector of $ V_o(t) = ( 250sin(\theta) , 250cos(\theta), 100 )$ $ V(t) = V_o(t) $
$$ V(t) = [ 250sin(t) , 250cos(t) , 100t ] $$
where you can convert the speed to radians per second and inject it into the t variable and you now can control the velocity that the object has while still on the path