Relative Motion: Rain Example

Q. Rain is falling vertically with a velocity of $15m{s}^{-1}$. A person rides a bicycle with a speed of $15m{s}^{-1}$ in the west to east direction. What is the direction (angle with vertical) in which he should hold his umbrella to save himself from the rain?

1. ${45}^{\circ }$
2. ${90}^{\circ }$
3. ${30}^{\circ }$
4. ${15}^{\circ }$

Q. Rain appears to fall vertically downwards to a man walking at the rate of $3\text{km/h}$. When he increases his speed to $6\text{km/h}$, the rain appears coming to him at an angle of ${45}^{\circ }$ with the vertical. The speed of rain is

1. $3\sqrt{3}\text{km/h}$
2. $3\sqrt{2}\text{km/h}$
3. $3\sqrt{5}\text{km/h}$
4. $5\sqrt{3}\text{km/h}$

Q. If $\overrightarrow{V_r}$ is the velocity of rain falling vertically and $\overrightarrow{V_m}$ is the velocity of a man walking on a level road, and $\theta$ is the angle with vertical at which he should hold the umbrella to protect himself, then the magnitude of relative velocity of rain w.r.t man is given by

1. $\sqrt{|\overrightarrow{V_r}{|}^2+|\overrightarrow{V_m}{|}^2}$
2. $\sqrt{|\overrightarrow{V_r}{|}^2+|\overrightarrow{V_m}{|}^2+2|\overrightarrow{V_r}||\overrightarrow{V_m}|\cos \theta }$
3. $\sqrt{|\overrightarrow{V_r}{|}^2+|\overrightarrow{V_m}{|}^2-2|\overrightarrow{V_r}||\overrightarrow{V_m}|\cos \theta }$
4. $\sqrt{|\overrightarrow{V_r}{|}^2-|\overrightarrow{V_m}{|}^2}$

Q. A stationary man observes that the rain is falling vertically downward. When he starts running with a velocity of $12\text{km/h}$, he observes that the rain is falling at an angle ${60}^{\circ }$ with the vertical. The actual velocity of rain is

1. $12\sqrt{3}\text{km/h}$
2. $6\sqrt{3}\text{km/h}$
3. $4\sqrt{3}\text{km/h}$
4. $2\sqrt{3}\text{km/h}$

Q.

At a certain place, horizontal component is $\sqrt{3}$ times the vertical component, the angle of dip at this place is.

1. $0$

2. $\frac{\mathrm{\pi }}{3}$

3. $\frac{\mathrm{\pi }}{6}$

4. none of these

Q. Rain is falling vertically downwards with a velocity of $3\text{km/hr}$. A man walks in the rain with a velocity of $4\text{km/hr}$. The rain drop will fall on the man with a velocity of

1. 5 km/hr
2. 4km/hr
3. 1km/hr
4. 3 km/hr

Q.

A man sitting in a bus traveling in a direction from West to East with a speed of $40$ km/hr observes that the raindrops are falling vertically down. To another man standing on the ground, the rain will appear

1. To fall vertically down

2. To fall at an angle going from West to East

3. To fall at an angle going from East to West

4. The information given is insufficient to decide the direction of rain

Q. A man running on a horizontal road at $8\text{km/h}$ finds the rain falling vertically. He increases his speed to $12\text{km/h}$ and finds that the drops make an angle ${30}^{\circ }$ with the vertical. Find the speed of the rain with respect to the road (in $\text{km/hr}$).

1. $2\sqrt{7}$
2. $4\sqrt{7}$
3. $5\sqrt{7}$
4. $6\sqrt{7}$

Q. A man standing on a road has to hold his umbrella at ${30}^{\circ }$ with the vertical to keep the rain away. He throws the umbrella and starts running at $10{\text{kmh}}^{-1}$ . He finds that raindrops are hitting his head vertically. The actual speed of raindrops is

1. $20{\text{kmh}}^{-1}$
2. $10\sqrt{2}{\text{kmh}}^{-1}$
3. $20\sqrt{3}{\text{kmh}}^{-1}$
4. $10{\text{kmh}}^{-1}$

Q. A man is standing on a road and observes that rain is falling at angle ${45}^{\circ }$ with the vertical. The man starts running on the road with constant acceleration $0.5{\text{m/s}}^2$. After a certain time from the start of the motion, it appears to him that rain is still falling at angle ${45}^{\circ }$ with the vertical, with speed $2\sqrt{2}\text{m/s}$. Motion of the man is in the same vertical plane in which the rain is falling. Then which of the following statement(s) are true:

1. It is not possible
2. Speed of the rain relative to the ground is $2\text{m/s}$
3. Speed of the man when he finds rain to be falling at angle ${45}^{\circ }$ with the vertical, is $4\text{m/s}$
4. The man has travelled a distance $16\text{m}$ on the road by the time he again finds rain to be falling at angle ${45}^{\circ }$

Q. Rain is falling vertically with a velocity of $15m{s}^{-1}$. A person rides a bicycle with a speed of $15m{s}^{-1}$ in the west to east direction. What is the direction (angle with vertical) in which he should hold his umbrella to save himself from the rain?

1. ${45}^{\circ }$
2. ${90}^{\circ }$
3. ${30}^{\circ }$
4. ${15}^{\circ }$

Q. Rain is falling vertically downwards with a velocity of $3\text{km/hr}$. A man walks in the rain with a velocity of $4\text{km/hr}$. The rain drop will fall on the man with a velocity of

1. 5 km/hr
2. 4km/hr
3. 1km/hr
4. 3 km/hr

Q. A man standing on a road has to hold his umbrella at ${30}^{\circ }$ with the vertical to keep the rain away. He throws the umbrella and starts running at $10km/h$. He finds that raindrops are hitting his head vertically. Find the speed of raindrops with respect to the ground.

1. $5km/h$
2. $10km/h$
3. $10\sqrt{2}km/h$
4. $20km/h$

Q.

From the previous problem at what angle to the horizontal must be steered in order to reach a point on the opposite bank directly east from the starting point? (Assuming his speed w.r.t. to the river is 4 m/s)

1. $0^{\circ }$

2. ${30}^{\circ }$

3. ${45}^{\circ }$

4. ${60}^{\circ }$

Q. A man moving with velocity $5\text{m/s}$ along a straight line observes rain falling vertically at the rate of $10\text{m/s}$. Find the speed of rain with respect to ground, and angle$\left(\theta \right)$ made from vertical by the rain as seen from ground.

1. $5\sqrt{5}\text{m/s},\theta =\frac{{\tan }^{-1}4}{2}$
2. $5\sqrt{5}\text{m/s},\theta ={\tan }^{-1}\frac{1}{2}$
3. $10\text{m/s},\theta ={\tan }^{-1}2$
4. $10\text{m/s},\theta ={\tan }^{-1}\frac{1}{2}$

Q.

Rain is falling vertically with a speed of $12m{s}^{-1}$.A cyclist is moving east to west with a speed of $12\sqrt{3}m{s}^{-1}$.In order to protect himself from rain at what angle he should hold his umbrella w.r.t vertical.

1. ${60}^{\circ }$

2. $0^{\circ }$

3. ${30}^{\circ }$

4. ${45}^{\circ }$

Q. The rain is falling vertically with a speed $4m{s}^{-1}$ and a man is moving due east with a speed $3m{s}^{-1}$. With what speed does the rain appear to be falling (for the man)? Also at what angle should he hold the umbrella to protect himself from rain?

1. $5m{s}^{-1},{37}^{\circ }\text{with respect to vertical}$
2. $5m{s}^{-1},{53}^{\circ }\text{with respect to vertical}$
3. $5m{s}^{-1},{45}^{\circ }\text{with respect to horizontal}$
4. $5m{s}^{-1},{30}^{\circ }\text{with respect to horizontal}$

Q.

A flag is hoisted on the steamer boat.On the basis of the data given below find out the direction of flutter of

the flag

${\overrightarrow{v}}_{steamer/water}=10\sqrt{3}m{s}^{-1}$ towards North

${\overrightarrow{v}}_{water}=5m{s}^{-1}$ towards East

${\overrightarrow{v}}_{wind}=5m{s}^{-1}$ towards East

(Assume the wind does not affect the steamer)

1. North

2. South

3. East

4. West

Q. A boat which has a speed of $5\text{m/s}$ in still water crosses the river of width $25\text{m}$ in 10 seconds. The boat is heading at an angle of $\alpha$ with downstream, where $\alpha$ is equal to

1. ${150}^{\circ }$
2. ${120}^{\circ }$
3. ${90}^{\circ }$
4. ${60}^{\circ }$

Q.

A man wishes to cross a river in a boat. If he crosses the river in minimum time he takes 10 minutes with a drift of 120 m. If he crosses the river taking shortest route, he takes 12.5 minutes, find velocity of the boat with respect to water.

1. $12m/min$

2. $20m/min$

3. $5m/min$

4. $31m/min$

Q. Rain driven by the wind, falls on a railway compartment with a velocity of $20\text{m/s}$, at an angle of ${30}^{\circ }$ to the vertical. The train moves, along the direction of wind flow, at a speed of $108\text{km/hr}$. Determine the apparent velocity ( in $\text{m/s}$ upto two decimal places) of rain as observed by a person sitting in the train.( Given $\sqrt{7}=2.64$)

Q. When a man moves down an inclined plane with a constant speed $5\text{m/s}$ which makes an angle of ${37}^{\circ }$ with the horizontal, he finds that the rain is falling vertically downwards. When he moves up the same inclined plane with the same speed, he finds that the rain makes an angle $\theta ={\tan }^{-1}\left(\frac{7}{8}\right)$ with the horizontal. The speed of the rain with respect to the ground is

1. $\sqrt{116}\text{m/s}$
2. $\sqrt{32}\text{m/s}$
3. $\sqrt{10}\text{m/s}$
4. $5\text{m/s}$

Q. To a man running at the rate of $4\text{km/h}$, the rain appears to fall vertically. When he increases his speed to $8\text{km/h}$, rain appears coming to man at an angle of ${45}^{\circ }$ with vertical. Find the magnitude of velocity of rain with respect to ground.

1. $3\sqrt{2}\text{km/h}$
2. $4\sqrt{2}\text{km/h}$
3. $2\sqrt{2}\text{km/h}$
4. $5\sqrt{2}\text{km/h}$

Q. A man holds an umbrella at ${30}^{\circ }$ with the vertical to keep himself dry. He then runs at a speed of $10\text{m/s}$ and finds the rain drops to be hitting vertically. Speed of the rain drops w.r.t the running man and w.r.t earth are

1. $20\text{m/s}$, $10\text{m/s}$
2. $10\text{m/s}$, $20\sqrt{3}\text{m/s}$
3. $10\sqrt{3}\text{m/s}$, $20\text{m/s}$
4. $20\text{m/s}$, $10\sqrt{3}\text{m/s}$

Q. A stationary person observes that rain is falling vertically down at $30\text{km/h}$. A cyclist is moving up on an inclined plane making an angle ${30}^{\circ }$ with horizontal at $10\text{km/h}$. In which direction should the cyclist hold his umbrella to prevent himself from the rain ?

1. At an angle ${\tan }^{-1}\left(\frac{3\sqrt{3}}{5}\right)$ with the inclined plane.
2. At an angle ${\tan }^{-1}\left(\frac{3\sqrt{3}}{5}\right)$ with the horizontal.
3. At an angle ${\tan }^{-1}\left(\frac{\sqrt{3}}{7}\right)$ with the inclined plane.
4. At an angle ${\tan }^{-1}\left(\frac{\sqrt{3}}{7}\right)$ with the vertical.

Q.

A glass wind screen whose inclination with the vertical can be changed is mounted on a car. The car moves horizontally with a speed of 2 m/s. At what angle $\alpha$ with the vertical should the screen be placed so that the rain drops falling downwards with velocity 6 m/s strike the windscreen perpendicularly?

1. $\tan -1\left(\frac{1}{3}\right)$

2. $\tan -1\left(3\right)$

3. $\cos -1\left(3\right)$

4. $\sin -1\left(\frac{1}{3}\right)$

Q. A man at rest observes that rain is falling vertically downwards. When he moves with a velocity of $3\text{km/h}$, he holds an umbrella at an angle of ${37}^{\circ }$ from the vertical to avoid rain. The velocity of the rain with respect to man is

1. $8\text{km/h}$
2. $7\text{km/h}$
3. $6\text{km/h}$
4. $5\text{km/h}$

Q. If $\overrightarrow{V_r}$ is the velocity of rain falling vertically and $\overrightarrow{V_m}$ is the velocity of a man walking on a level road, and $\theta$ is the angle with vertical at which he should hold the umbrella to protect himself, then the magnitude of relative velocity of rain w.r.t man is given by

1. $\sqrt{|\overrightarrow{V_r}{|}^2-|\overrightarrow{V_m}{|}^2}$
2. $\sqrt{|\overrightarrow{V_r}{|}^2+|\overrightarrow{V_m}{|}^2}$
3. $\sqrt{|\overrightarrow{V_r}{|}^2+|\overrightarrow{V_m}{|}^2+2|\overrightarrow{V_r}||\overrightarrow{V_m}|\cos \theta }$
4. $\sqrt{|\overrightarrow{V_r}{|}^2+|\overrightarrow{V_m}{|}^2-2|\overrightarrow{V_r}||\overrightarrow{V_m}|\cos \theta }$

Q. A man standing on a road has to hold his umbrella at ${30}^{\circ }$ with the vertical to keep the rain away. He throws the umbrella and starts running at $10km/h$. He finds that raindrops are hitting his head vertically. Find the speed of raindrops with respect to the ground.

1. $5km/h$
2. $10km/h$
3. $10\sqrt{2}km/h$
4. $20km/h$

Q. The rain is falling vertically with a speed $4m{s}^{-1}$ and a man is moving due east with a speed $3m{s}^{-1}$. With what speed does the rain appear to be falling (for the man)? Also at what angle should he hold the umbrella to protect himself from rain?

1. $5m{s}^{-1},{37}^{\circ }\text{with respect to vertical}$
2. $5m{s}^{-1},{53}^{\circ }\text{with respect to vertical}$
3. $5m{s}^{-1},{45}^{\circ }\text{with respect to horizontal}$
4. $5m{s}^{-1},{30}^{\circ }\text{with respect to horizontal}$