Question

A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are$4.00m$to the north, $2.00m$
northeast, and $1.00m$at$30.0°$west of south (Fig. P3.27). Starting at the same initial point, an expert golfer could make the hole in what single displacement?

Short answer

The required single displacement is$4.64mat78.6°$ .

Step-by-step solution

Define single displacement and magnitude

1. A single displacement reaction, also known as a single replacement reaction, is a type of oxidation-reduction chemical process in which an ion or element moves out of a molecule and is replaced by another.
2. In simple terms, magnitude means 'distance or number.' It depicts the absolute or relative size or direction in which an object moves in the feeling of motion.

Determine the single displacement

We must find the displacements in the x and y axes individually in order to find the single displacement, that is, the resultant displacement of the three strokes.

$$\begin{gathered} x=0+\left(2\times cos45°\right)-\left(1\times sin30°\right) \\ =0.914m \\ y=4+\left(2\times sin45°\right)-\left(1\times cos30°\right) \\ =4.55m \end{gathered}$$

Determine the magnitude

As a result, the magnitude is determined by

$$\begin{gathered} \left|\overrightarrow{R}\right|=\sqrt{x^2+y^2} \\ \left|\overrightarrow{R}\right|=\sqrt{0.{914}^2+4.{55}^2} \\ \left|\overrightarrow{R}\right|=4.64m \end{gathered}$$

In addition, the angle is determined by

$$\begin{gathered} \theta =ta{n}^{-1}\frac{y}{x} \\ =ta{n}^{-1}\frac{4.55}{0.914} \\ =78.6^0 \end{gathered}$$

Hence, the required single displacement is$4.64mat78.6°$ .