Mario, a hockey player, is skating due south at a speed of 4.79 m/s relative to the ice. A teammate passes the puck to him. The puck has a speed of 9.13 m/s and is moving in a direction of 17.8 ° west of south, relative to the ice. What are (a) the magnitude and (b) direction (relative to due south) of the puck's velocity, as observed by Mario

Respuesta :

Mario's velocity vector relative to the ice is

[tex]\vec v_{M/I}=-4.79\,\vec\jmath[/tex]

(note that all velocities mentioned here are given in m/s)

The puck is moving in a direction of 17.8 degrees west of south, or 252.2 degrees counterclockwise relative to east. Its velocity vector relative to the ice is then

[tex]\vec v_{P/I}=9.13(\cos252.2^\circ\,\vec\imath+\sin252.2^\circ\,\vec\jmath)=-2.79\,\vec\imath-8.69\,\vec\jmath[/tex]

The velocity of the puch relative to Mario is

[tex]\vec v_{P/M}=\vec v_{P/I}+\vec v_{I/M}=\vec v_{P/I}-\vec v_{M/I}[/tex]

[tex]\vec v_{P/M}=-2.79\,\vec\imath-3.90\,\vec\jmath[/tex]

Then, relative to Mario,

a. the puck is traveling at a speed of [tex]\boxed{\|\vec v_{P/M}\|=4.80}[/tex], and

b. is moving in a direction [tex]\theta[/tex] such that

[tex]\tan\theta=\dfrac{-3.90}{4.80}\implies\theta=-126^\circ[/tex]

which is about [tex]\boxed{35.6^\circ}[/tex] west of south.