Parametrize trajectory of a hocjey puck

[plexus0208]

Homework Statement

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).

Homework Equations

general eqn: (Rcosθ, Rsinθ), where R is the radius of the puck

The Attempt at a Solution

radius = R = 1
frequency = f = 2
angular frequency = w = 2πf = 4π
θ = wt

(cos4πt + 10cos(π/2)t, sin4πt + 10sin(π/2)t)
Is this answer right?

 

[HallsofIvy]

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)).

Add those two motions.

 

[plexus0208]

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.