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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.

u=126 km/h=126×518m/s=35m/s
v=0
s=200m
Newton's Equation of motion
v2−u2=2as
02−352=2a(200)
a=−3.0625m/s2
Also
v=u+at
0=35−3.06t
t=11.4s