To determine the angle at which a man should hold an umbrella to protect himself from the rain, we need to consider the relative motion between the man and the rain.
### Step-by-Step Solution:
1. Given Information:
- The man is moving due east with a velocity of [tex]\(5 \, \text{m/s}\)[/tex].
- The rain is falling vertically downwards with a velocity of [tex]\(4 \, \text{m/s}\)[/tex].
2. Understanding Relative Motion:
The man needs to hold the umbrella in such a way that it counteracts the relative direction of the rain, providing protection. This involves determining the resultant vector of the man's horizontal motion and the vertical motion of the rain.
3. Formulating the Problem:
- Let [tex]\( \mathbf{v}_\text{east} \)[/tex] represent the man's velocity due to east, which has a magnitude of [tex]\( 5 \, \text{m/s} \)[/tex].
- Let [tex]\( \mathbf{v}_\text{vertical} \)[/tex] represent the rain's vertical velocity, which has a magnitude of [tex]\( 4 \, \text{m/s} \)[/tex].
4. Determining the Direction:
- The resultant velocity vector [tex]\( \mathbf{v}_\text{resultant} \)[/tex] can be found using the components of [tex]\( \mathbf{v}_\text{east} \)[/tex] and [tex]\( \mathbf{v}_\text{vertical} \)[/tex]. The angle [tex]\( \theta \)[/tex] between the resultant vector and the vertical axis can be determined using trigonometry.
- Specifically, we use the tangent function:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\mathbf{v}_\text{east}}{\mathbf{v}_\text{vertical}} = \frac{5}{4}
\][/tex]
5. Calculating the Angle:
To find the angle [tex]\( \theta \)[/tex], we take the arctangent of [tex]\( \frac{5}{4} \)[/tex]:
[tex]\[
\theta = \tan^{-1}\left(\frac{5}{4}\right)
\][/tex]
6. Converting to Radians and Degrees:
- The angle [tex]\( \theta \)[/tex] in radians is approximately [tex]\( 0.896 \)[/tex] radians.
- To convert radians to degrees:
[tex]\[
\theta \times \frac{180}{\pi} \approx 51.34^\circ
\][/tex]
### Conclusion:
The man should hold the umbrella at an angle of approximately [tex]\(0.896\)[/tex] radians or [tex]\(51.34^\circ\)[/tex] to the vertical in the anticlockwise direction to protect himself from the vertically falling rain while he moves eastwards.
