Find all possible curvatures with respect to time given a specific speed

[Meziel]

This problem is not from school but it is something I'm trying to solve for a computer program. Given an exact speed S I am trying to find find a position equation using time where I can randomize x and y, and then solve for z.

So far this is what I have:
I'm representing a vector in 3 space by < V_x(t), V_y(t), V_z(t)>
the magnitude of a velocity vector is the speed which = sqrt( V_x(t)^{2 +} V_y(t)^{2 +} V_z(t)²)
because my velocity is constant
S = sqrt( V_x(t)^{2 +} V_y(t)^{2 +} V_z(t)²)
Now I need to put this into a parametric form so solving for the V_x(t) I get
V_x(t) = sqrt( S² - V_y(t)^{2 -} V_z(t)² )

Now here is the hard part. This is what I can't do. I just started calc3 and calc2 did not teach me this so here it is: I need the equation for R_x(t) so the obvious thing to do is to take the integral/anti-derivative.
so....

the integral of V_x(t) = R_x(t) = integral sqrt( S² - V_y(t)^{2 -} V_z(t)² ) dt = ?

If I have done anything wrong in my problem thus far or if there is another way of doing this please let me know. Also, If you don't understand anything also tell me and I'll try to clarify. I sent a picture of the formula I'm trying to solve if that helps.

 

[Steven G]

This problem is not from school but it is something I'm trying to solve for a computer program. Given an exact speed S I am trying to find find a position equation using time where I can randomize x and y, and then solve for z.

So far this is what I have:
I'm representing a vector in 3 space by < V_x(t), V_y(t), V_z(t)>
the magnitude of a velocity vector is the speed which = sqrt( V_x(t)^{2 +} V_y(t)^{2 +} V_z(t)²)
because my velocity is constant
S(t) = sqrt( V_x(t)^{2 +} V_y(t)^{2 +} V_z(t)²)
Now I need to put this into a parametric form so solving for the V_x(t) I get
V_x(t) = sqrt( S² - V_y(t)^{2 -} V_z(t)² ) It looks like you are saying that S is a constant but S is a function of t. So sqrt( S² - V_y(t)^{2 -} V_z(t)² ) is V_x(t)

Now here is the hard part. This is what I can't do. I just started calc3 and calc2 did not teach me this so here it is: I need the equation for R_x(t) so the obvious thing to do is to take the integral/anti-derivative.
so....

the integral of V_x(t) = R_x(t) = integral sqrt( S² - V_y(t)^{2 -} V_z(t)² ) dt = ? First simplify sqrt( S² - V_y(t)^{2 -} V_z(t)² ) to V_x(t) which really means you dd not get anywhere

If I have done anything wrong in my problem thus far or if there is another way of doing this please let me know. Also, If you don't understand anything also tell me and I'll try to clarify. I sent a picture of the formula I'm trying to solve if that helps.

Wait a minute. Although it has been a while since I looked at this material I think that I see where you are going wrong.

V(t)=< V_x(t), V_y(t), V_z(t)>
So R(t)= < R_x(t), R_y(t), R_z(t)>

If you know what V_x(t) is then you take the integral of it w/r/t t and you'll get R_x(t).

For example if V(t) = (3t^2, 6t, 7/t> then R_x(t) =t^3 (the integral of 3t^2 dt)

 

[Meziel]

I declared S as the magnitude of the vector |v(t)| because I want the speed to stay constant. It was a given in the problem I am solving. I have V_x(t) and that is like i said the sqrt( S² - V_y(t)² - V_z(t)² ).
I didn't rewrite the whole velocity vector but it is this:

V(t) = <sqrt( S² - V_y(t)² - V_z(t)² ), sqrt( S² - Vx(t)² - V_z(t)² ), sqrt( S² - Vx(t)² - Vy(t)² )>

So like you said integrating V_x(t) isn't easy because i'm having to integrate sqrt( S² - V_y(t)² - V_z(t)² ) dt where S is a constant. That is my problem.

 

[Steven G]

I declared S as the magnitude of the vector |v(t)| because I want the speed to stay constant. It was a given in the problem I am solving. I have V_x(t) and that is like i said the sqrt( S² - V_y(t)² - V_z(t)² ).
I didn't rewrite the whole velocity vector but it is this:

V(t) = <sqrt( S² - V_y(t)² - V_z(t)² ), sqrt( S² - Vx(t)² - V_z(t)² ), sqrt( S² - Vx(t)² - Vy(t)² )>

So like you said integrating V_x(t) isn't easy because i'm having to integrate sqrt( S² - V_y(t)² - V_z(t)² ) dt where S is a constant. That is my problem.

i do not see how S(t)= sqrt( V_x(t)^{2 +} V_y(t)^{2 +} V_z(t)²) is constant as it is a (non constant) function of t. If t=6 units or t =7 units why do you think that s(6) = s(7) = some constant?

 

[Meziel]

i do not see how S(t)= sqrt( V_x(t)^{2 +} V_y(t)^{2 +} V_z(t)²) is constant as it is a (non constant) function of t. If t=6 units or t =7 units why do you think that s(6) = s(7) = some constant?

The reason |V(t)| is constant is because i want to traverse through an arbitrary curve at a constant speed. It's a given in the problem.

 

[Steven G]

The reason |V(t)| is constant is because i want to traverse through an arbitrary curve at a constant speed. It's a given in the problem.

Suppose V=<3t, 5t^3, 4>. Then S^2 = (3t)^2 + (5t^3)^2 + (4)^2 which clearly is not constant.

You say that your speed, s, is constant. Fine. But how does this happen? If V = < V_x(t), V_y(t), V_z(t)> what must be true so that |V| is constant?

 

[Meziel]

You say that your speed, s, is constant. Fine. But how does this happen? If V = < V_x(t), V_y(t), V_z(t)> what must be true so that |V| is constant?

the sqrt( V_x(t)^{2 +} Vy(t)^{2 +} Vz(t)² ) = S

--------------------- below this line is not related to the problem but is an example of what i'm going for
it's like saying in a function f(t) = 4x where f(t) = a constant let's say 8
so 8 = 4x so x is 2

--------------------- back to the problem
so in |V(t)| at every interval that t increases from 0->infinity |V(t)| will always equal a constant... in the case of the problem i called it S. it could be 2, 3, 4, whatever.
so with the formula |V(t)| = sqrt( V_x(t)^{2 +} Vy(t)^{2 +} Vz(t)² ) that i derived you can change this to S = sqrt( V_x(t)^{2 +} Vy(t)^{2 +} Vz(t)² )

 

[Meziel]

Suppose V=<3t, 5t^3, 4>. Then S^2 = (3t)^2 + (5t^3)^2 + (4)^2 which clearly is not constant.

Also the way you used this isn't right. S doesn't replace V(t) it replaces the magnitude of V(t) which is |V(t)|. you can find this through Pythagorean theorem.

 

[Steven G]

Also the way you used this isn't right. S doesn't replace V(t) it replaces the magnitude of V(t) which is |V(t)|. you can find this through Pythagorean theorem.

You wrote the magnitude of a velocity vector is the speed which = sqrt( V_x(t)^{2 +} V_y(t)^{2 +} V_z(t)²). You found S and I found S^2. If S is constant so is S^2

 

[Meziel]

You wrote the magnitude of a velocity vector is the speed which = sqrt( V_x(t)^{2 +} V_y(t)^{2 +} V_z(t)²). You found S and I found S^2. If S is constant so is S^2

I believe you are right. Even though I'm setting the magnitude as a constant it is still a function of time and needs to be integrated as such. therefore if i integrate i get 1/3|R(t)|³ = 1/3R_x(t)^3 + 1/3Ry(t)^3 + 1/3Rz(t)^3
solving for x i get R_x(t) = (|R(t)|^{3 -} Ry(t)³ - Rz(t)³)^1/3 which makes sense because if i set |V(t)| = 30 then |R(t)| = 30t. if i set Ry(t) = 0 and Rz(t) = 0 then solve for x i get 30t. The velocity of <30t, 0, 0> = <30, 0, 0> so the magnitude is 30. Thank you so much for your help.

 

[Ishuda]

This problem is not from school but it is something I'm trying to solve for a computer program. Given an exact speed S I am trying to find find a position equation using time where I can randomize x and y, and then solve for z.

So far this is what I have:
I'm representing a vector in 3 space by < V_x(t), V_y(t), V_z(t)>
the magnitude of a velocity vector is the speed which = sqrt( V_x(t)^{2 +} V_y(t)^{2 +} V_z(t)²)
because my velocity is constant
S = sqrt( V_x(t)^{2 +} V_y(t)^{2 +} V_z(t)²)
Now I need to put this into a parametric form so solving for the V_x(t) I get
V_x(t) = sqrt( S² - V_y(t)^{2 -} V_z(t)² )

Now here is the hard part. This is what I can't do. I just started calc3 and calc2 did not teach me this so here it is: I need the equation for R_x(t) so the obvious thing to do is to take the integral/anti-derivative.
so....

the integral of V_x(t) = R_x(t) = integral sqrt( S² - V_y(t)^{2 -} V_z(t)² ) dt = ?

If I have done anything wrong in my problem thus far or if there is another way of doing this please let me know. Also, If you don't understand anything also tell me and I'll try to clarify. I sent a picture of the formula I'm trying to solve if that helps.

Hi,
Just out of curiosity, and if you don't mind answering the question, do you/will you have a functional form or you doing this with a numerical integration on a time step (or some other) basis?

 

[Meziel]

Hi,
Just out of curiosity, and if you don't mind answering the question, do you/will you have a functional form or you doing this with a numerical integration on a time step (or some other) basis?

I believe I got the answer already. < ((St)³ - Ry(t)³ - Rz(t)³)^1/3 + C, ((St)³ - Rx(t)³ - Rz(t)³)^1/3 + C, ((St)³ - Rx(t)³ - Ry(t)³)^1/3​ + C> Where S is a constant speed.

 

[Ishuda]

I declared S as the magnitude of the vector |v(t)| because I want the speed to stay constant. It was a given in the problem I am solving. I have V_x(t) and that is like i said the sqrt( S² - V_y(t)² - V_z(t)² ).
I didn't rewrite the whole velocity vector but it is this:

V(t) = <sqrt( S² - V_y(t)² - V_z(t)² ), sqrt( S² - Vx(t)² - V_z(t)² ), sqrt( S² - Vx(t)² - Vy(t)² )>

So like you said integrating V_x(t) isn't easy because i'm having to integrate sqrt( S² - V_y(t)² - V_z(t)² ) dt where S is a constant. That is my problem.

You have to be careful here. What you have gives a constant |V| for any three functions V_x(t), V_y(t), and V_z(t), however it does not necessarily give the proper components of the velocity vector. For example, consider the following functions
V_x = 2 t
V_y = t^1/2
V_z = 1
S = 10
Note: The V_x(t), V_y(t), and V_z(t) are NOT the components of V(t). What is V(t)?

Oh, and don't forget, there is also a sign choice on square root.