State True or False: 
Adding a scalar to a vector of the same dimensions is a meaningful algebraic operation.

A. FALSE
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A person standing on a road has to hold his umbrella at 300 with the vertical to keep the rain away. He throws the umbrella and starts running at 10 m/s. He finds that raindrops are hitting his head vertically. Find the speed of raindrops with respect to, 

a) the road,
b) the moving person.


When a person is at rest with respect to ground, the rain is coming to him at an angle 30o with the vertical, i.e., along OB. 

As the person moves along OA with velocity 10 m/s, the relative velocity of rain w.r.t person is along OC as shown in fig. below.

                        

Here, o 

Therefore, velocity of rain w.r.t ground = straight v with rightwards harpoon with barb upwards on top subscript r g end subscript space equals space stack O B with rightwards harpoon with barb upwards on top 

Velocity of person w.r.t. ground = straight v with rightwards harpoon with barb upwards on top subscript p g end subscript space equals space stack O A with rightwards harpoon with barb upwards on top comma space w h e r e space O A space equals space 10 space m divided by s space equals space C B

Velocity of rain w.r.t person = straight v with rightwards harpoon with barb upwards on top subscript r p end subscript space equals space stack O C with rightwards harpoon with barb upwards on top 

a) increment OCB comma space OB space equals space CB divided by space sin space 30 to the power of 0

space space space space space space space space space space space space space space space space space space space space space equals space 10 divided by space left parenthesis 1 divided by 2 right parenthesis thin space equals space 20

Therefore comma space velocity space of space rain space straight w. straight r. straight t space ground space equals space 20 space straight m divided by straight s

b)

In increment space OCB comma space OB space equals space CB divided by tan space 30 to the power of straight o 

                          = 10/ (1/square root of 3) = 10square root of 3 

Therefore, velocity of train w.r.t person = 10square root of 3 m/s  

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The blades of an aeroplane propeller are rotating at the rate of 600 revolutions per minute. Calculate its angular velocity. 

Here, 

Frequency, straight nu space equals space 600 space revolutions space per space minute 

                  equals 600 over 60 revolutions per second 

We know, 

Angular frequency, straight omega = 2 πν space equals space 2 space straight pi space straight x space 600 over 60 
                              = 20 straight pi rad sec-1 

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A police jeep is chasing a culprit going on a motorbike. The motorbike crosses a turning at a speed of 72 km/hr. the jeep follows it at a speed of 108 km/hr; crossing the turning 10 seconds later than the bike. Assuming that they travel at a constant speeds, how far from the turning will the jeep catch up with the bike. 

Speed of the motorbike  = 72 km/hr  = 20 m/s 

Speed of police jeep = 108 km/hr = 30 m/s 

Let the jeep catch up with the bike after t seconds. 

Therefore, 

       30 t = 20 ( t + 10)


rightwards double arrow        t = 20 sec 

Distance travelled by jeep in 20 sec, 

         s =30 x 20 = 600 m 

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state True or False: 
Adding of any two scalars is a meaningful algebraic operation.

A. FALSE
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