Relative Motion: Rain Example

Q. A man walks at the rate of $3\text{km/hr}$. Rain appears to him falling in vertical direction at the rate of $3\sqrt{3}\text{km/hr}$. Which of the following options is correct regarding the actual velocity of rain?

1. $3\text{km/hr}$, inclined at an angle of ${30}^{\circ }$ to the vertical towards the man's motion.
2. $6\text{km/hr}$, inclined at an angle of ${45}^{\circ }$ to the vertical towards the man's motion.
3. $6\text{km/hr}$, inclined at an angle of ${60}^{\circ }$ to the vertical towards the man's motion.
4. $6\text{km/hr}$, inclined at an angle of ${30}^{\circ }$ to the vertical towards the man's motion.

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

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

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. 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. A man running on a horizontal road at $8{\text{ms}}^{-1}$ finds rain falling vertically. If he increases his speed to $12{\text{ms}}^{-1}$, he finds that drops make ${30}^{\circ }$ angle with the vertical. Find the velocity of rain with respect to the road.

1. $8{\text{ms}}^{-1}$
2. $4\sqrt{7}{\text{ms}}^{-1}$
3. $8\sqrt{2}{\text{ms}}^{-1}$
4. $7\sqrt{3}{\text{ms}}^{-1}$

Q.

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

1. $12\text{m/min}$

2. $20\text{m/min}$

3. $5\text{m/min}$

4. $31\text{m/min}$

Q. Rain is falling vertically downwards with a velocity of $4km/h$. A man walks in the rain with a velocity of $3km/h$. The raindrops will fall on the man with a velocity of magnitude

1. $1km/h$
2. $3km/h$
3. $4km/h$
4. $5km/h$

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 person standing on a road has to hold his umbrella at 60${}^{\circ }$ with the vertical to keep away rain. He throws the umbrella and starts running in horizontal direction at a speed of $20\text{m/s}$. He finds that the rain drops are hitting his head vertically. Find the speed of rain drops with respect to ground.

1. $\frac{20}{\sqrt{3}}\text{m/s}$
2. $17\text{m/s}$
3. $\frac{40}{\sqrt{3}}\text{m/s}$
4. $11.5\text{m/s}$

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

Q. A boy aims a gun at a bird from a point, at a horizontal distance of 100 m. If the gun can impart a velocity of 500 $m{s}^{-1}$ to the bullet. At what height above the bird must he aim his gun in order to hit it (take g = 10 $m{s}^{-2}$)

1. 10 cm
2. 50 cm
3. 100 cm
4. 20 cm

Q. Rain is falling vertically with a speed of $30m/s$. A woman rides a bicycle with a speed of $12m/s$ in east to west direction. What is the direction in which she should hold her umbrella?

1. At an angle of ${\tan }^{-1}(\frac{2}{5})$ with the vertical towards the east
2. At an angle of ${\tan }^{-1}(\frac{2}{5})$ with the vertical towards the west
3. At an angle of ${\tan }^{-1}(\frac{5}{2})$ with the vertical towards the east
4. At an angle of ${\tan }^{-1}(\frac{5}{2})$ with the vertical towards the west

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

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. A ship $A$ is moving eastwards with a speed of $20\text{km/h}$ and a ship $B$ , $50\text{km}$ south of $A$ is moving northwards with a speed of $20\text{km/h}$. The time after which the distance between ships become shortest is

1. $1\text{h}$
2. $3.5\text{h}$
3. $1.25\text{h}$
4. $2.5\text{h}$

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

1. $3\text{kmph}$
2. $4\text{kmph}$
3. $1\text{kmph}$
4. $5\text{kmph}$

Q. A man is coming down an inclined plane of angle ${30}^{\circ }$ with speed $2\sqrt{3}\text{m/s}$. He has to keep his umbrella vertical to protect himself from rain. If the actual magnitude of velocity of rain is $5\text{m/s}$, at what angle with vertical should he keep his umbrella when he is at rest?

1. ${45}^{\circ }$
2. ${37}^{\circ }$
3. ${53}^{\circ }$
4. ${30}^{\circ }$

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. When a car is at rest, its driver sees raindrops falling on it vertically. When driving the car with speed $v,$ he sees that raindrops are coming at an angle ${60}^{\circ }$ from the horizontal. On further increasing the speed of the car to $(1+\beta )v,$ this angle changes to ${45}^{\circ }$. The value of $\beta$ is close to :

1. $0.50$
2. $0.41$
3. $0.37$
4. $0.73$

Q. A man is walking due east at the speed of $3\text{km/h}$. Rain appears to fall down vertically at the rate of $3\text{km/h}$. The magnitude of actual velocity of the rain is

1. $3\sqrt{2}\text{km/h}$
2. $3\text{km/h}$
3. $6\text{km/h}$
4. $9\text{km/h}$

Q. A man moving with a velocity of $5\text{m/s}$ on a horizontal road observes that raindrops fall at an angle of ${45}^{\circ }$ with the vertical. When he moves with a velocity of $16\text{m/s}$ along an inclined plane, which is inclined at ${30}^{\circ }$ with the horizontal, he observes raindrops falling vertically downward as shown in the figure. Find the actual velocity ${\overrightarrow{V}}_r$ of the raindrops.

1. $\overrightarrow{V_r}=8\sqrt{3}\hat{i}-(8\sqrt{3}-5)\hat{j}$
2. $\overrightarrow{V_r}=4\sqrt{3}\hat{i}-(4\sqrt{3}-5)\hat{j}$
3. $\overrightarrow{V_r}=2\sqrt{3}\hat{i}-(2\sqrt{3}-5)\hat{j}$
4. $\overrightarrow{V_r}=\sqrt{3}\hat{i}-(\sqrt{3}-5)\hat{j}$

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. 1km/hr
3. 3 km/hr
4. 4km/hr

Q. Rain pouring down at an angle $\alpha$ with the vertical has a speed of $10\text{m/s}$. A girl runs against the rain with the speed of $8\text{m/s}$ and sees that the rain makes an angle $\beta$ with the vertical, then relation between $\alpha$ and $\beta$ is

1. $\tan \alpha =\frac{8+10\sin \beta }{10\cos \beta }$
2. $\tan \beta =\frac{8+10\sin \alpha }{10\cos \alpha }$
3. $\tan \alpha =\tan \beta$
4. $\tan \alpha =\cot \beta$

Q. A car is moving in a rainstorm at a speed of $36\text{km/hr}$. The rain drops fall vertically w.r.t the ground with a constant speed of $20\text{m/s}$. The speed of the raindrops w.r.t the car is

1. $10\sqrt{5}\text{m/s}$
2. $50\sqrt{5}\text{m/s}$
3. $5\sqrt{5}\text{m/s}$
4. $10\sqrt{2}0\text{m/s}$

Q. It is raining vertically with a velocity of $25{\text{ms}}^{-1}$. A woman rides a bicycle with a speed of $10{\text{ms}}^{-1}$ in north to south direction. What is the direction (angle with vertical) in which she should hold her umbrella to save herself from the rain?

1. ${\tan }^{-1}\left(0.4\right)$
2. ${\tan }^{-1}\left(1\right)$
3. ${\tan }^{-1}\left(\sqrt{3}\right)$
4. ${\tan }^{-1}\left(2.6\right)$

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

Q. Rain, driven by the wind, falls on a railway compartment with a velocity of $20m/s$, at an angle of ${30}^{\circ }$ to the vertical. The train moves, along the direction of wind flow, at a speed of $108km/hr$. Determine the apparent velocity of rain for a person sitting in the train.

1. $20\sqrt{7}m/s$
2. $15\sqrt{7}m/s$
3. $12\sqrt{7}m/s$
4. $10\sqrt{7}m/s$

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

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