Question 3.9: Translating Magnitude and Direction into Component Form Jai ......
Translating Magnitude and Direction into Component Form
Jai alai (pronounced “high lie”) is a court game in which players use a long hand-shaped basket strapped to their wrist to propel a ball (Fig. 3.37). The object of the game is to bounce the ball against the wall in such a way that an opponent cannot catch and return the ball. Suppose a jai alai ball collides with a wall at a speed of 69 m/s and a 55° angle from the vertical as shown in Figure 3.38. Choose an x – y coordinate system and write the ball’s velocity in component form.
INTERPRET and ANTICIPATE
Choose a coordinate system and draw a sketch (Fig 3.39). For convenience, place the tail of \overrightarrow{ v } at the origin and draw the vector components of \overrightarrow{ v }. In anticipation of using \theta=\tan ^{-1}\left(A_y / A_x\right) (Eq. 3.14), we also show the angle \theta measured counterclockwise from the x axis. The velocity vector points in the negative x and y directions, so we expect a numerical result of the form \vec{v}=(- ____ \hat{ \imath } – ____ \hat{ \jmath }) m / s.
SOLVE
Solve for \theta .
Trigonometry (Eqs. 3.9 and 3.10)
A_x=A \cos \theta \quad \quad (3.9) \\ A_y=A \sin \theta \quad \quad (3.10)gives the scalar components.
\begin{aligned}& v_x=v \cos \theta=(69 m / s ) \cos 215^{\circ}=-57 m / s \\& v_y=v \sin \theta=(69 m / s ) \sin 215^{\circ}=-40 m / s\end{aligned}(two significant figures)
To write the ball’s velocity in component form, place the scalar components in front of the appropriate unit vectors.
\overrightarrow{ v }=(-57 \hat{\imath}-40 \hat{\jmath}) m / sCHECK and THINK
Both components are negative as we expected. Breaking a velocity into its horizontal and vertical scalar components is an important part of analyzing two-dimensional motion. So, let’s use the acute 55° angle and trigonometry to find the components a second time. The velocity vector is the hypotenuse of a right triangle; the x component is opposite the angle, and the y component is adjacent. We can use the sine and cosine functions to find the magnitude of the vector components.
The negative signs do not come automatically using this method. Instead, we must use our sketch to reason that both components are negative.
