Question

Being part of the “Gators,” the University of Florida golfing team must play on a putting green with an alligator pit. Figure 3-22 shows an overhead view of one putting challenge of the team; an xy coordinate system is superimposed. Team members must putt from the origin to the hole, which is at xy coordinates (8 m, 12 m), but they can putt the golf ball using only one or more of the following displacements, one or more times:$\stackrel{\mathbf{\rightharpoonup }}{{\mathbf{d}}_{\mathbf{1}}}\mathbf{=}\left(8m\right)\hat{\mathbf{i}}\mathbf{+}\left(6m\right)\hat{\mathbf{j}}\mathbf{,}\stackrel{\mathbf{\rightharpoonup }}{{\mathbf{d}}_{\mathbf{2}}}\mathbf{}\mathbf{=}\left(6m\right)\hat{\mathbf{j}}\mathbf{,}\mathbf{}\stackrel{\mathbf{\rightharpoonup }}{{\mathbf{d}}_{\mathbf{3}}}\mathbf{}\mathbf{=}\left(8m\right)\hat{\mathbf{i}}$The pit is at coordinates (8 m, 6 m). If a team member putts the ball into or through the pit, the member is automatically transferred to Florida State University, the arch rival. What sequence of displacements should a team member use to avoid the pit and the school transfer?

Short answer

The sequence will be$\stackrel{\rightharpoonup }{d_2,}\stackrel{\rightharpoonup }{d_1}or\stackrel{\rightharpoonup }{d_2,}\stackrel{\rightharpoonup }{d_3}$

Step-by-step solution

Given information

The displacements vector are

$$\begin{gathered} \stackrel{\rightharpoonup }{{\mathrm{d}}_1}=\left(8\mathrm{m}\right)\hat{\mathrm{i}}+\left(6\mathrm{m}\right)\hat{\mathrm{j}} \\ \stackrel{\rightharpoonup }{{\mathrm{d}}_1}=\left(6\mathrm{m}\right)\hat{\mathrm{j}} \\ \stackrel{\rightharpoonup }{{\mathrm{d}}_3}=\left(8\mathrm{m}\right)\hat{\mathrm{i}} \end{gathered}$$

The pit is at coordinates (8 m, 6 m).

To understand the concept

If the two vectors are directed exactly in the same direction, then they can be added using the simple addition rules. But if their directions are different, then we have to use the components of vectors along unit vectors and add them. Using the vector addition laws, we can add any two vectors.

Formula:

$\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{b}=\stackrel{\rightharpoonup }{c}$

To find the sequence of displacements

The ball is at origin $\left(0,0\right)$. We have to put ball in the hole at $\left(8,12\right)$. We can use displacements given as ,$\stackrel{\rightharpoonup }{d_1},\stackrel{\rightharpoonup }{d_2}\mathrm{and}\stackrel{\rightharpoonup }{{\mathrm{d}}_3}$. The ball should not go inside the pit at ( 8, 6 ). The displacements of ball should be multiples of $\stackrel{\rightharpoonup }{d_1},\stackrel{\rightharpoonup }{d_2}\mathrm{and}\stackrel{\rightharpoonup }{{\mathrm{d}}_3}$.

First, the ball goes along $\stackrel{\rightharpoonup }{d_2}$and then $\stackrel{\rightharpoonup }{d_1}$, by using vector addition law.

$\stackrel{\rightharpoonup }{d}=\stackrel{\rightharpoonup }{d_2}+\stackrel{\rightharpoonup }{{\mathrm{d}}_1}$

Substitute the given values of the vectors.

$$\begin{gathered} \stackrel{\rightharpoonup }{\mathrm{d}}=\left(6\mathrm{m}\right)\hat{\mathrm{j}}+\left(8\mathrm{m}\right)\hat{\mathrm{i}}+\left(6\mathrm{m}\right)\hat{\mathrm{j}} \\ \stackrel{\rightharpoonup }{\mathrm{d}}=\left(8\mathrm{m}\right)\hat{\mathrm{i}}+\left(12\mathrm{m}\right)\hat{\mathrm{j}} \end{gathered}$$

By using this sequence, the ball can be put in the hole at (8, 12 ) .

Another way the ball goes is along , ${\stackrel{\rightharpoonup }{d}}_2,{\stackrel{\rightharpoonup }{d}}_2\mathrm{and}\mathrm{then}{\stackrel{\rightharpoonup }{\mathrm{d}}}_3$.

According to the vector addition law,

$\stackrel{\rightharpoonup }{d}={\stackrel{\rightharpoonup }{d}}_2+{\stackrel{\rightharpoonup }{d}}_2+{\stackrel{\rightharpoonup }{d}}_3$

Substitute the given values of the vectors.

$$\begin{gathered} \stackrel{\rightharpoonup }{\mathrm{d}}=\left(6\mathrm{m}\right)\hat{\mathrm{j}}+\left(6\mathrm{m}\right)\hat{\mathrm{j}}+\left(8\mathrm{m}\right)\hat{\mathrm{i}} \\ \stackrel{\rightharpoonup }{\mathrm{d}}=\left(8\mathrm{m}\right)\hat{\mathrm{i}}+\left(6\mathrm{m}\right)\hat{\mathrm{j}} \end{gathered}$$

By using this sequence, ball can be put into the hole at (8,12 ).

Thus, the sequence will be${\stackrel{\rightharpoonup }{d}}_2,{\stackrel{\rightharpoonup }{d}}_1\mathrm{or}{\stackrel{\rightharpoonup }{\mathrm{d}}}_2,{\stackrel{\rightharpoonup }{\mathrm{d}}}_2,{\stackrel{\rightharpoonup }{\mathrm{d}}}_3$