# Thread: Velocities of curves.

1. ## Velocities of curves.

Calculate the velocities v1,v2 of alpha1, alpha2 respectively at p = (1,0,0).

alpha1 = (-Pi, Pi) -> Reals(3), defined by alpha1(t) = (cost,sint,t)

alpha2 = (-Pi, Pi) -> R(Reals)(3), defined bby alpha2(t) = (cost, sint, -t).

2. $\displaystyle \alpha_1(t)=(cos(t), \; sin(t), \; t)$
$\displaystyle \alpha_2(t)=(cos(t), \; sin(t), \; -t)$
$\displaystyle p=(1, \; 0, \; 0)$
These are expressions for the position of a particle at a time t.

Since $\displaystyle p=(1, \; 0, \; 0)$ we need a t value. We can find this by:

$\displaystyle (1, \; 0, \; 0)=(cos(t), \; sin(t), \; t) \Rightarrow \ t=0$.

We only have expressions for position. Therefore we need to differentiate them to have an expression for the velocity.

$\displaystyle \alpha_1(t)=(cos(t), \; sin(t), \; t)$
$\displaystyle \bigtriangledown \alpha_1(t)=(-sin(t), \; cos(t), \; 1)$

$\displaystyle \alpha_2(t)=(cos(t), \; sin(t), \; -t)$
$\displaystyle \bigtriangledown \alpha_2(t)=(-sin(t), \; cos(t), \; -1)$

(Can someone post back and tell me if this is notationally correct. I'm not sure if I should be using $\displaystyle \bigtriangledown$ here.)

When t=0:

$\displaystyle \bigtriangledown \alpha_1(t)=(-sin(0), \; cos(0), \; 1)=(0, \; 1, \; 1)$
$\displaystyle \bigtriangledown \alpha_2(t)=(-sin(0), \; cos(0), \; -1)=(0, \; 1, \; -1)$