On a certain day, rain was falling vertically with a speed 30 m/s. The wind starts blowing after some time with a speed 10 m/s in N to S direction. In which direction should a boy, waiting at bus stop, hold his umbrella to save himself from rain?

To the boy waiting at bus stop, the rain will appear to fall in the direction of the resultant of rain and wind.
Let θ be the angle which the resultant velocity subtends with the vertical.

To the boy waiting at bus stop, the rain will appear to fall in the d
From the figure
       space space space space space space tanθ equals straight v subscript straight w over straight v subscript straight r equals 10 over 30 equals 1 third
                  straight theta equals 18 degree 26 apostrophe
∴   Boy should hold the umbrella at angle 18 degree 26 apostrophe North of vertical.

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Two trains, each of length 100 m, are running on parallel tracks. One overtakes the other in 20 seconds and one crosses the other in 10 second. Calculate the velocities of each train? 


Let u and v be the velocity of two trains A and B. While overtaking the relative velocity of train A w.r.t B = u -v. 

While crossing, the relative velocity of train A w.r.t B = u+ v.

Total distance to be travelled by car A while crossing = 100 + 100 = 200 

So, 

space space space space space space space space space space 20 space equals space fraction numerator 200 over denominator straight u space minus space straight v end fraction space

rightwards double arrow space space straight u space minus straight v space equals space 10

Also comma space

space space space space space space space space space space space space space 10 space equals space fraction numerator 200 over denominator straight u space plus space straight v end fraction

rightwards double arrow space space space space straight u plus straight v space equals space 20 space

On space solving comma space we space get

straight u space equals space 15 space straight m divided by straight s comma space is space the space velocity space of space first space train comma

straight v space equals space 5 space straight m divided by straight s comma space is space the space velocity space of space the space second space train. space

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Rain is falling vertically with speed of 30 m/s. A woman rides a bicycle with a speed of 10 m/s in N to S direction. In which direction she should hold the umbrella?

Let stack straight v subscript straight r with rightwards arrow on top be the velocity of rain and <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
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#6 {main}</pre> be the velocity of the woman. To the woman the rain will appear to come from the direction of stack straight v subscript straight r with rightwards arrow on top minus straight v with rightwards arrow on top. Therefore, woman should hold the umbrella in the direction of left parenthesis stack straight v subscript straight r with rightwards arrow on top minus straight v with rightwards arrow on top right parenthesis which is given by
            space space space space tanθ equals straight v over straight v subscript straight r equals 10 over 30 equals 1 third

Let  be the velocity of rain and  be the velocity of the woman. T
i.e. she should hold the umbrella at an angle straight theta equals tan to the power of negative 1 end exponent left parenthesis 1 divided by 3 right parenthesis south of vertical.

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Wind is blowing west to east along two parallel tracks. Two trains moving with the same speed in opposite directions have the same steam tracks, if one is double than the other, what is the speed of each train? 


Let v and w be the velocity of each train and wind respectively. 

According to the question, we have

(v+w) = 2 (v-w)


rightwards double arrow space straight v space equals space 3 straight w 

That is, velocity of each train is 3 times the velocity of wind. 

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A body travelling along a straight line traversed one-third of the total distance with a velocity 4 m/s. The remaining part of the distance was covered with a velocity 6 m/s for the other half of time. What is the mean velocity averaged over the whole time of motion? 

Let s be the total distance travelled by body.

If t1 is the time taken to cover first one-third distance, then 


               straight t subscript 1 space equals space fraction numerator straight s divided by 3 over denominator 4 end fraction space equals space straight s over 12 space
Let t2 be the time for each of the remaining two journeys. 

Then, 


                 fraction numerator 2 straight s over denominator 3 end fraction space equals space 2 t subscript 2 space plus space 6 t subscript 2 space
space space space space space space space space equals space 8 t subscript 2 


                    straight t subscript 2 space equals space fraction numerator 2 straight s over denominator 3 space straight x 8 end fraction space equals space straight s over 12 space 
Therefore, 

Average velocity = fraction numerator Total space displacement over denominator total space time end fraction 

                         = fraction numerator straight s over denominator straight t subscript 1 space plus space 2 straight t subscript 2 end fraction space equals space fraction numerator s over denominator left parenthesis s divided by 12 right parenthesis space plus space 2 left parenthesis s divided by 12 right parenthesis end fraction space equals space 4 space m divided by s

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