# Parametrize trajectory of a hocjey puck

1. Nov 16, 2009

### plexus0208

1. The problem statement, all variables and given/known data
A hockey puck of radius 1 slides along the ice at a speed 10 in the direction of the vector (1,1). As it slides, it spins in a counterclockwise direction at 2 revolutions per unit time. At time t = 0, the puck’s center is at the origin (0,0).

Find the parametric equations for the trajectory of the point P on the edge of the puck initially at (1,0).

2. Relevant equations
general eqn: (Rcosθ, Rsinθ), where R is the radius of the puck

3. The attempt at a solution
frequency = f = 2
angular frequency = w = 2πf = 4π
θ = wt

(cos4πt + 10cos(π/2)t, sin4πt + 10sin(π/2)t)

Last edited: Nov 16, 2009
2. Nov 16, 2009

### HallsofIvy

Staff Emeritus
Re: Parametrize

No, it isn't.

First, follow the motion of the center of the puck. It is, at time t= 0, at (0, 0) and moves in the direction of vector <1, 1> with speed 10: (x, y)= (vt, vt) which has speed $\sqrt{v^2+ v^2}= v\sqrt{2}= 10$. $v= 10/\sqrt{2}= 5\sqrt{2}$. The center of the puck, at time t, is at (5\sqrt{2}t, 5/sqrt{2}t).

Now look at the rotation. Since it makes two revolutions per unit time, it makes one revolution when t= 1/2. $(x, y)= (cos(\omega t), sin(\omega t))$ and has period 1/2: $\omega (1/2)= 2\pi$ so $\omega= 4\pi$. [itex](x,y)= (cos(4\pi t), sin(4\pi t)).

3. Nov 16, 2009

### plexus0208

Re: Parametrize

Woops, I had a typo. Instead of (cos4πt + 10cos(π/2)t, sin4πt + 10sin(π/2)t), I meant:
(cos4πt + 10cos(π/4)t, sin4πt + 10sin(π/4)t)

Here, 10cos(π/4)t and 10sin(π/4)t are equivalent to 5sqrt2.