Question

Rain is falling vertically with velocity 80 cm/s.

(a) How should you hold your umbrella? (b) You start walking towards the east with velocity 60 cm/s. How should you hold your umbrella? (c) You are walking towards the west with velocity 60 cm/s. How should you hold your umbrella? (d) You are walking towards the north with velocity 60 cm/s. How should you hold your umbrella? (e) You are walking towards the south with velocity 80 cm/s. How should you hold your umbrella? 42. When you are standstill in rain, you have to hold your umbrella vertically to protect yourself.

Answer 1

The umbrella should be held such that its axis is aligned with the velocity of rain relative to the observer (the person holding the umbrella).

Answer 2

The umbrella should be held such that its axis is aligned with the velocity of rain relative to the observer (the person holding the umbrella). Let $\vec{v}_r$ be the velocity of rain relative to the ground and $\vec{v}_m$ be the velocity of the man relative to the ground. The velocity of rain relative to the man is given by $\vec{v}_{rm} = \vec{v}_r - \vec{v}_m$.

Given: Rain is falling vertically with velocity 80 cm/s. So, $\vec{v}_r = (0, -80)$ cm/s (taking the vertical direction as y-axis, with negative indicating downwards).

(a) How should you hold your umbrella? This implies the observer is standing still. $\vec{v}_m = (0, 0)$ cm/s. $\vec{v}_{rm} = \vec{v}_r - \vec{v}_m = (0, -80) - (0, 0) = (0, -80)$ cm/s. The rain appears to fall vertically downwards. Answer: The umbrella should be held vertically.

(b) You start walking towards the east with velocity 60 cm/s. How should you hold your umbrella? Let East be the positive x-direction. $\vec{v}_m = (60, 0)$ cm/s. $\vec{v}_{rm} = \vec{v}_r - \vec{v}_m = (0, -80) - (60, 0) = (-60, -80)$ cm/s. Let $\theta$ be the angle the relative velocity makes with the vertical. The horizontal component of $\vec{v}_{rm}$ is $V_{rm,x} = -60$ cm/s (towards West). The vertical component of $\vec{v}_{rm}$ is $V_{rm,y} = -80$ cm/s (downwards). $\tan \theta = \frac{|V_{rm,x}|}{|V_{rm,y}|} = \frac{|-60|}{|-80|} = \frac{60}{80} = \frac{3}{4}$. $\theta = \tan^{-1}(3/4) = 37^\circ$. Since the horizontal component of the relative velocity is towards the West, the umbrella must be tilted towards the East (the direction of walking) to block the rain. Answer: The umbrella should be held at an angle of $37^\circ$ with the vertical, tilted towards the East.

(c) You are walking towards the west with velocity 60 cm/s. How should you hold your umbrella? Let West be the negative x-direction. $\vec{v}_m = (-60, 0)$ cm/s. $\vec{v}_{rm} = \vec{v}_r - \vec{v}_m = (0, -80) - (-60, 0) = (60, -80)$ cm/s. The horizontal component of $\vec{v}_{rm}$ is $V_{rm,x} = 60$ cm/s (towards East). The vertical component of $\vec{v}_{rm}$ is $V_{rm,y} = -80$ cm/s (downwards). $\tan \theta = \frac{|V_{rm,x}|}{|V_{rm,y}|} = \frac{|60|}{|-80|} = \frac{60}{80} = \frac{3}{4}$. $\theta = \tan^{-1}(3/4) = 37^\circ$. Since the horizontal component of the relative velocity is towards the East, the umbrella must be tilted towards the West (the direction of walking) to block the rain. Answer: The umbrella should be held at an angle of $37^\circ$ with the vertical, tilted towards the West.

(d) You are walking towards the north with velocity 60 cm/s. How should you hold your umbrella? Let North be the positive x-direction (using a 2D plane for horizontal and vertical). $\vec{v}_m = (60, 0)$ cm/s. $\vec{v}_{rm} = \vec{v}_r - \vec{v}_m = (0, -80) - (60, 0) = (-60, -80)$ cm/s. The horizontal component of $\vec{v}_{rm}$ is $V_{rm,x} = -60$ cm/s (towards South). The vertical component of $\vec{v}_{rm}$ is $V_{rm,y} = -80$ cm/s (downwards). $\tan \theta = \frac{|V_{rm,x}|}{|V_{rm,y}|} = \frac{|-60|}{|-80|} = \frac{60}{80} = \frac{3}{4}$. $\theta = \tan^{-1}(3/4) = 37^\circ$. Since the horizontal component of the relative velocity is towards the South, the umbrella must be tilted towards the North (the direction of walking) to block the rain. Answer: The umbrella should be held at an angle of $37^\circ$ with the vertical, tilted towards the North.

(e) You are walking towards the south with velocity 80 cm/s. How should you hold your umbrella? Let South be the negative x-direction. $\vec{v}_m = (-80, 0)$ cm/s. $\vec{v}_{rm} = \vec{v}_r - \vec{v}_m = (0, -80) - (-80, 0) = (80, -80)$ cm/s. The horizontal component of $\vec{v}_{rm}$ is $V_{rm,x} = 80$ cm/s (towards North). The vertical component of $\vec{v}_{rm}$ is $V_{rm,y} = -80$ cm/s (downwards). $\tan \theta = \frac{|V_{rm,x}|}{|V_{rm,y}|} = \frac{|80|}{|-80|} = \frac{80}{80} = 1$. $\theta = \tan^{-1}(1) = 45^\circ$. Since the horizontal component of the relative velocity is towards the North, the umbrella must be tilted towards the South (the direction of walking) to block the rain. Answer: The umbrella should be held at an angle of $45^\circ$ with the vertical, tilted towards the South.