What is the Relationship between Arc Length and Angle Phi on a Helix?

[sitzpillow]

Dear physicist,
my task is to calculate the acceleration of a particle of mass m which moves without friction in the Earth's gravitational field on a helix:
The helix is parameterized as shown:

$$x(\phi)=a cos \phi$$
$$y(\phi)=a sin \phi$$
$$z(\phi)=c \phi$$

formed with a radius a,gradient c as constants and the angle $$\phi$$ which mimics the projection of the radius vector on the x, y plane of the axis x with $$\phi \in 0<= \phi<\infty$$

What is the relationship between the arc length s and the angle phi?
Also, I need to derivate the tangents, normals and binormal vector by using $$\overrightarrow{r}(s)$$
and calculate the end nor the path velocity (with s (t = 0) = 0, s' (0) = 0).

I'm afraid not to have any approaches to solve the problem :/
I would appreciate every hint.

 

[PeroK]

Dear physicist,
my task is to calculate the acceleration of a particle of mass m which moves without friction in the Earth's gravitational field on a helix:
The helix is parameterized as shown:

$$x(\phi)=a cos \phi$$
$$y(\phi)=a sin \phi$$
$$z(\phi)=c \phi$$

formed with a radius a,gradient c as constants and the angle $$\phi$$ which mimics the projection of the radius vector on the x, y plane of the axis x with $$\phi \in 0<= \phi<\infty$$

What is the relationship between the arc length s and the angle phi?
Also, I need to derivate the tangents, normals and binormal vector by using $$\overrightarrow{r}(s)$$
and calculate the end nor the path velocity (with s (t = 0) = 0, s' (0) = 0).

I'm afraid not to have any approaches to solve the problem :/
I would appreciate every hint.

Can you think of a conserved quantity that might make this problem easier?

You may also want to consider velocity expressed in other than Cartesian coordinates (if that isn't giving too much help!).

 

[berkeman]

@sitzpillow -- You are required to show us your efforts toward the solution before we can offer much tutorial help. Please use the hint provided by PeroK and show us your efforts...