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The phenomenon of scattering of light helps to explain the blue color of sky

1. Scattering of light takes place when size of the scattering object (such as air molecules) is very very small as compared to wavelength of light
2. According to Rayleigh's law of scattering, the amount of scattering is inversely proportional to λ⁴.
3. Thus when light from sun travels through earth's atmosphere, the different wavelengths get scattered from their path through different amounts obeying Rayleigh's law of scattering.
4. Since wavelength of blue color is approximately half the wavelength of red color, the scattering of blue light is about 2⁴ times more than red light.
5. Due to this blue color predominates and the sky appears blue.

Given, the actual path length travelled is 23 km, displacement is 10 km and the time taken is 28 min.

(a)

Average speed of the taxi is given as,

v avg = Total Path length Time

Here, s is the total path length

Substitute the values in above equation.

v avg = Total Path length Time = 23 km ( 28 min )( 1 h 60 min ) =49.3  km/h

Hence, the average speed of the taxi is 49.3  km/h .

(b)

Magnitude of the average velocity is given as,

v avg = Displacement Time

Substitute the values in the above equation.

v avg = Displacement Time = 10 km ( 28 min )( 1 h 60 min ) =21.4  km/h

(c)

From the calculation in (a) and (b), it is obtained that the average speed is not equal to the magnitude of average velocity.

Here, Volume of water =30 m3
t=15 min=15×60 s=900 s;Height,h=40 m
Efficiency, η=30%
Density of water =103kg m−3
∴ Mass of water pumped =Volume×Density
=(30 m3)(103 kg m−3)=3×104kg
Poutput=Wt=mght=(3×104kg)(10 ms−2)(40 m)900 s
=43×104W
Efficiency, η=PoutputPinput
Pinput=Poutputη=4×1043×30100=49×105
=44.4×103W=44.4 kW.

Linear momentum of particle, →p=px^i+py^j+pz^k

Mass of the big structure, M = 50,000 kg
Inner radius of the column, r = 30 cm = 0.3 m
Outer radius of the column, R = 60 cm = 0.6 m
Young’s modulus of steel, Y=2×1011Pa

Total force exerted, F=Mg=50000×9.8N

Stress = Force exerted on a single column =50000×9.84=122500N
Young’s modulus, Y=StressStrain

Strain =(F/A)Y
Where,
Area, A=π(R2−r2)=π((0.6)2−(0.3)2)

Strain =122500/[π((0.6)2−(0.3)2)×2×1011]=7.22×10−7

Hence, the compressional strain of each column is 7.22×10−7.

Water is flowing at a rate of 3.0 litre/min.
The geyser heats the water, raising the temperature from 27oC to 77oC.
Initial temperature, T1=27oC
Final temperature, T2=77oC
Rise in temperature,T=T1−T2=77−27=50oC
Heat of combustion =4×104J/g
Specific heat of water, C=4.2J/goC
Mass of flowing water, m= 3.0 litre/min = 3000 g/min
Total heat used, Q=mcT
=3000×4.2×50
=6.3×105J/min
Rate of consumption =6.3×105/(4×104) = 15.75 g/min.