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Rain is falling vertically with a speed of 30 m s-1. A woman rides a bicycle with a speed of 10 m s-1 in the north to south direction. What is the direction in which she should hold her umbrella?

Let, 

vc = Velocity of the cyclist, and

vr = Velocity of falling rain 

The woman must hold her umbrella in the direction of the relative velocity (v) of the rain, in order to protect herself from the rain. 

straight v space equals space straight v subscript straight r space end subscript plus space left parenthesis negative straight v subscript straight c right parenthesis space

space space space equals space 30 space plus space left parenthesis negative 10 right parenthesis space

space space space equals space 20 space straight m divided by straight s space

tan space straight theta space equals fraction numerator space straight v subscript straight c over denominator space straight v subscript straight r space end subscript end fraction space equals fraction numerator space 10 space over denominator 30 end fraction
rightwards double arrow space straight theta space equals space tan to the power of negative 1 end exponent space open parentheses fraction numerator 1 space over denominator 3 end fraction close parentheses space
space space space space space space space space space equals tan to the power of negative 1 end exponent space left parenthesis 0.333 right parenthesis space almost equal to space 18 to the power of 0 space


Therefore, the woman must hold the umbrella towards the south, at an angle of 18o.

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An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10s apart is 30°, what is the speed of the aircraft?


The aircraft takes 10s to go from A to B and AB subtends an angle of 30° at O.
In right angled triangle OAC,
AC = OC tan 15° = 3400 x 0·2679 = 911m

The aircraft takes 10s to go from A to B and AB subtends an angle of
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Now, the speed of aircraft is,
space space straight v equals AB over straight t equals 1822 over 10 equals 182.2 space straight m divided by straight s

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Given a + b+ c + d = 0, which of the following statements
are correct :

(a) a, b, c, and d must each be a null vector,

(b) The magnitude of (a + c) equals the magnitude of
( b + d),

(c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d,

(d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear ?


a)

Incorrect

In order to make vectors a + b + c + d = 0, it is not necessary to have all the four given vectors to be null vectors. There are many other combinations which can give the sum zero.

(b)

Correct

a + b + c + d = 0

a + c = – (b + d)

Taking modulus on both the sides, we get,

| a + c | = | –(b + d)| = | b + d |

Hence, the magnitude of (a + c) is the same as the magnitude of (b + d).

(c)

Correct

a + b + c + d = 0

a = - (b + c + d)

Taking modulus both sides, we get,

| a | = | b + c + d |

| a | ≤ | a | + | b | + | c | ... (i)

Equation (i) shows that the magnitude of a is equal to or less than the sum of the magnitudes of b, c, and d.

Hence, the magnitude of vector a can never be greater than the sum of the magnitudes of b, c, and d.

(d)

Correct

For a + b + c + d = 0

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On an open ground, a motorist follows a track that turns to his left by an angle of 60after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.


The path followed by the motorist is a regular hexagon with side 500 m, as shown in the given figure.

Let the motorist start from point P.

The motorist takes the third turn at S.

∴ Magnitude of displacement = PS = PV + VS = 500 + 500 = 1000 m

Total path length = PQ + QR + RS = 500 + 500 +500 = 1500 m

The motorist takes the sixth turn at point P, which is the starting point.

∴ Magnitude of displacement = 0

Total path length = PQ + QR + RS + ST + TU + UP

                      = 500 + 500 + 500 + 500 + 500 + 500

                      = 3000 m

The motorist takes the eight turn at point R.

∴ Magnitude of displacement = PR

 
Therefore, the magnitude of displacement is 866.03 m at an angle of 30° with PR.

This implies AC makes an angle 30° with the initial direction.

Total path length = 8 × 500 = 4000 m. 

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Read each statement below carefully and state, with reasons and examples, if it is true or false:
A scalar quantity is one that
(a) is conserved in a process.

(b) can never take negative values.

(c) must be dimensionless.

(d) does not vary from one point to another in space.
(e) has the same value for observers with different orientations of axes.

(a) False -  When we carry out the adiabatic process, the scalar quantity temperature does not conserve.

In inelastic collisions, the K.E. (scalar quantity) does not conserve.

(b) False - scalar quantity can have negative value e.g. .

If the angle between  is greater than then W will be negative e.g. work done by gravitational force when body is raised upward is negative.

(c) False -  Scalar quantity may have dimensions, e.g. Dimensional formula of work is [M1L2T-2].

(d) False - Temperature is a scalar quantity and it changes as we go higher, the gravitational potential is a scalar quantity and it varies with the height.

(e) True.
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