-
State for each of the following physical quantities, if it is a scalar or a vector : volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.
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Pick out the two scalar Quantities in the following list: force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, reaction as per Newton's third law, relative velocity.
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Pick out the only vector Quantity in the following list:
Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
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State with reasons, whether the following algebraic operations with scalar and vector physical Quantities are meaningful:
(a) Adding any two scalars.
(b) Adding a scalar to a vector of the same dimensions.
(c) Multiplying any vector by any scalar.
(d) Multiplying any two scalars.
(e) Adding any two vectors.
(f) Adding a component of a vector to the same vector.
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Read each statement below carefully and state with reasons, if it is true or false :
(a) The magnitude of a vector is always a scalar.
(b) Each component of a vector is always a scalar.
(c) The total path length is always equal to the magnitude of the displacement vector of a particle.
(d) The average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time.
(e) Three vectors not lying in a plane can never add up to give a null vector.
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Establish the following vector
inequalities geometrically or otherwise:
(a)
(b)
(c)
(d)
When does the equality sign above apply
?
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Given
which of
the following statements are correct:
(a)
and
must each
be a null vector,
(b) The magnitude of
equals
the magnitude of
(c) The magnitude of a can never
be greater than the sum of the magnitudes of
and
(d)
must lie
in the plane of
and
if a and
are not collinear, and in the line of
and
, they are
collinear ?
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Three girls skating on a circular ice
ground of radius 200 m start from a point P on the edge of the ground and reach
a point Q diametrically opposite to P following different paths as shown in Fig.
4.101. What is the magnitude of the displacement vector for each?
For which girl is this equal to
the actual length of path skated ?
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A cyclist travels from centre O of a
circular park of radius 1 km and reaches point P. After cycling along 1/4th of
the circumference along
he
returns to the centre of the park along
. If the
total time taken is 10 minutes, calculate (i) net displacement (ii) average
velocity and (ii) average speed of the cyclist.
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On an open ground, a motorist follows a track that turns to his left by an angle of 60° after 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.
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A passenger arriving in a new town wishes to go from the station to a hotel located 10km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 minutes. What is (i) the average speed of the taxi and (ii) the magnitude of average velocity ? Are the two equal ?
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Rain is falling vertically with a speed
of
. A woman
rides a bicycle with a speed of
in the
north to south direction. What is the relative velocity of rain with respect to
the woman ? What is the direction in which she should hold her umbrella to
protect herself from the rain?
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A man can swim with a speed of
in still
water. How long does he take to cross the river 1 km wide, if the river flows
steadily at
and he
makes his strokes normal to the river current ? How far from the river does he
go, when he reaches the other bank ?
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In a harbour, wind is blowing at the
speed of
and the
flag on the mast of a boat anchored in the harbour flutters along the N-E
direction. If the boat starts moving at a speed of
to the
north, what is the direction of the flag on the mast of the boat ?
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The ceiling of a long hall is 25 m high.
What is the maximum horizontal distance that a ball thrown with a speed of
can go
without hitting the ceiling of the hall?
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A cricketer can throw a ball to a maximum horizontal distance of 100m, How high above the ground can the cricketer throw the same ball ?
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A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 seconds, what is the magnitude and direction of acceleration of the stone?
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An aircraft executes a horizontal loop
of radius 1km with a steady speed of
. Compare
its centripetal acceleration with the acceleration due to gravity.
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Read each statement below carefully and state, with reasons, if it is true or false:
(a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre.
(b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point.
(c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector.
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The position of a particle is given by
where t is in seconds and the
coefficients have the proper units for
to be in
metres.
(a) Find the v and a of the particle,
(b) What is the magnitude and direction of velocity of the particle at t = 2 s
? [Delhi 10]
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A particle starts from the origin at t =
0 s with a velocity of
and
moves in the
plane
with a aslant acceleration of
. (a) At
what time is the x-coordinate of the particle 16m ? What is the y-coordinate of
the particle at the time ? (b) What is the speed of the particle at the time ?
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(a) If
and
are unit
vectors along X- and Y-axis respectively, then what is the magnitude and direction
of
and
?
(b) Find the components of
along
the directions of vectors
and
.
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For any arbitrary motion in space, which
of the following relations are true :
(a)
(b)
(c)
(d)
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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.
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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 10 s apart is 30°, what is the speed of the aircraft ?
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A vector has magnitude and direction.
Does it have a location in space ? Can it vary with time ? Will two equal vectors
and
at
different locations in space necessarily have identical physical effects ? Give
examples in support of your answer.
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A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector ? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector ?
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Can you associate vectors with (a) the length of a wire bent into a loop, (b) a plane area, (c) a sphere ? Explain.
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A bullet fired at an angle of 30° with the horizontal hits the ground 3 km away. By adjusting the angle of projection, can one hope to hit a target 5 km away ? Assume the muzzle speed to be fixed and neglect air resistance.
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A fighter plane flying horizontally at
an altitude of 1.5 km with a speed
passes
directly overhead an antiaircraft gun. At what angle from the vertical should
the gun be fired for the shell muzzle speed
to hit
the plane ? At what maximum altitude should the pilot fly the plane to avoid
being hit ? Take
.
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A cyclist is riding with a speed of
. As he approaches
a circular turn on the road of radius 80 m, he applies brakes and reduces his
speed at the constant rate
.
What is the magnitude and direction of
the net acceleration of the cyclist on the circular turn ?
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(a) Show that for a projectile the angle
between the velocity and the X-axis as a function of time is given by
(b) Show that the projection
angle
for a
projectile launched from the origin is given by:
where the symbols have their usual
meaning.
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question_answer33)
The angle between \[\vec{A}=\hat{i}+\hat{j}\]
and \[\vec{B}\,=\,\hat{i}-\hat{j}\,\] is
(a) \[{{45}^{o}}\] (b) \[{{90}^{o}}\]
(c) \[-\,{{45}^{o}}\] (d) \[{{180}^{o}}\]
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question_answer34)
Which one of the
following statements is true?
(a) A scalar quantity
is the one that is conserved in a process.
(b) A scalar
quantity is the one that can never take negative values.
(c) A scalar
quantity is the one that does not vary from one point to another in space.
(d) A scalar
quantity has the same value for observers with different orientations of the
axes.
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question_answer35)
Figure
shows die orientation of two vectors \[u\] and v in the XY plane.
If \[u=\,a\hat{i}\,+\,b\hat{j}\]
and \[v=\,p\hat{i}+\,q\hat{j}\]
which of the following is
correct?
(a) a and p are
positive while b and q are negative.
(b) a, p and b are positive
while q is negative.
(c) a, q and b are positive
while p is negative.
(d) a, b, p and q are all
positive.
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question_answer36)
The component of a
vector r along X-axis will have maximum value if
(a) r is along positive
Y-axis
(b) r is along positive
X-axis
(c) r makes an angle of 45°
with the x-axis
(d) r is along negative
Y-axis
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question_answer37)
The horizontal
range of a projectile fired at an angle of \[{{15}^{o}}\] is 50 m. If it is
fired with the same speed at an angle of \[{{45}^{o}}\], its range will be
(a) 60 m (b)
71 m
(c) 100 m (d)
141 in
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question_answer38)
Consider
the quantities, pressure, power, energy, impulse, gravitational potential,
electrical charge, temperature, area. Out of these, the only vector quantities
are
(a) Impulse, pressure and
area
(b) Impulse and area
(c) Area and gravitational
potential
(d) Impulse and pressure
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question_answer39)
In
a two dimensional motion, instantaneous speed \[{{\upsilon }_{0}}\] is a positive
constant. Then which of the following are necessarily true?
(a) The average velocity is
not zero at any time.
(b) Average acceleration
must always vanish.
(c) Displacements
in equal time intervals are equal.
(d) Equal path
lengths are traversed in equal intervals.
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question_answer40)
In
a two dimensional motion, instantaneous speed \[{{\upsilon }_{0}}\] is a
positive coolant. Then which of the following are necessarily true?
(a) The acceleration of the
particle is zero.
(b) The acceleration of the
particle is bounded.
(c) The
acceleration of the particle is necessarily in the plane of motion.
(d) The particle
must be undergoing a uniform circular motion.
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question_answer41)
The vectors A, B
and C add up to zero. Find which is false.
(a) \[A\times B\times C\] is
not zero unless B, C are parallel.
(b) \[(A\times
B).C\] is not zero unless B, C are parallel.
(c) If A, B, C
define a plane, \[(A\times B)\times C\] is in that plane
(d) \[(A\times \,B)\,.\,C=\,|A|\,|B|\,+|C|\,\to
\,{{C}^{2}}\,={{A}^{2}}+{{B}^{2}}\]
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question_answer42)
It
is found that \[|\vec{A}+\vec{B}|\,=\,|\vec{A}|\]. This necessarily implies.
(a) B = 0
(b) A, B are antiparallel
(c) A, B are perpendicular
(d) A.B \[\le \] 0
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question_answer43)
Two
particulars are projected in air with speed \[{{\upsilon }_{0}}\] at angle \[{{\theta
}_{1}}\] and
\[{{\theta }_{2}}\] (both acute) to the horizontal, respectively. If the height
reached by the first particle is greater than that of the second, then tick the
right choices
(a) angle of projection : \[{{\theta
}_{1}}>{{\theta }_{2}}\]
(b) time of flight : \[{{T}_{1}}>\,{{T}_{2}}\]
(c) horizontal range : \[{{R}_{1}}>{{\upsilon
}_{2}}\]
(d) total energy : \[{{U}_{1}}>\text{
}{{U}_{2}}\]
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question_answer44)
A
particle slide down a frictionless parabolic \[(y={{x}^{2}})\] track \[(A-B-C)\]
starting from rest at point A (fig.) Point B is at the vertex of parabola and
point C is at a height less than that of point A. After C, the particle moves
freely in air as a projectile. If the particle reaches highest point at P, then
(a) KE at P = KE at B
(b) height at P = height at
A
(c) total energy at P =
total energy at A
(d) time of travel
from A to B = time of travel from B to P.
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question_answer45)
Following are four
different relations about displacement, velocity and acceleration for the
motion of a particle in general. Choose the incorrect one (s)
(a) \[{{\upsilon
}_{av}}=\,\frac{1}{2}\,[\upsilon \,({{t}_{1}})\,+\,\upsilon \,({{t}_{2}})]\]
(b) \[{{\upsilon
}_{av}}=\,\frac{r\,({{t}_{2}})-\,r({{t}_{1}})}{{{t}_{2}}-{{t}_{1}}}\]
(c) \[{{v}_{a\upsilon
}}=\frac{1}{2}(v({{t}_{2}})-v({{t}_{1}})({{t}_{2}}-{{t}_{1}})\]
(d) \[{{a}_{a\upsilon
}}=\,\frac{\nu \,({{t}_{2}})\,-\,\nu \,({{t}_{1}})}{{{t}_{2}}-{{t}_{1}}}\]
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question_answer46)
For
a particle performing uniform circular motion, choose the correct statement (s)
from the following:
(a) Magnitude of
particle velocity (speed) remains constant.
(b) Particle
velocity remains directed perpendicular to radius vector.
(c) Direction of
acceleration keep changing as particle moves.
(d) Angular
momentum is constant in magnitude but direction keeps changing.
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question_answer47)
For two vectors \[\vec{A}\]
and \[\vec{B},\,\,|\vec{A}+\,\vec{B}|\] \[=|\vec{A}-\vec{B}|\]
is always
true when
(a) \[=|\vec{A}|\,=\,|\vec{B}|\,\ne
\,0\]
(b) \[\vec{A}\,\,\bot
\,\,\vec{B}\]
(c) \[|\vec{A}|\,=\,|\vec{B}|\,\,\ne
\,0\] and
A and B are parallel or anti parallel
(d) when either \[|\vec{A}|\]
or \[\,|\vec{B}|\]
is zero.
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question_answer48)
A cyclist starts
from centre O of a circular park of radius 1 km and moves along the path OPRQO
as shown Fig. If he maintains constant speed of \[10\,m{{s}^{-1}},\] what is his
acceleration at point R in magnitude and direction?
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question_answer49)
A
particle is projected in air at some angle to the horizontal, moves along
parabola as shown in Fig. where x and y indicate horizontal and verticle
directions, respectively. Show in the diagram, direction of velocity of
acceleration at points A, B and C.
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question_answer50)
A
ball is thrown from a roof top at an angle of \[{{45}^{o}}\] above the
horizontal. It hits the ground a few seconds later. At what point during its
motion, does the ball have
(a) greatest speed
(b) smallest speed
(c) greatest acceleration ?
Explain
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question_answer51)
A football is
kicked into the air vertically upwards. What is its (a) acceleration, and (b)
velocity at the highest point?
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question_answer52)
\[\vec{A},\,\,\vec{B}\]
and \[\vec{C}\]
are three
non-collinear, non vectors. What can you say about direction of \[\vec{A}\times
\,(\vec{B}\times \,\vec{C})\]?
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question_answer53)
A
boy travelling in an open car moving on a levelled road with constant speed
tosses a ball vertically up in die air and catches it back. Sketch the motion
of the ball as observed by a boy standing on the footpath. Give explanation to
support your diagram.
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question_answer54)
A
boy throws a ball in air at \[{{60}^{o}}\] to the horizontal along a road with a speed
of 10 m/s (36 km/h). Another boy sitting in passing by car observes the ball.
Sketch the motion of the ball as observed by the boy in the car, if car has a
speed of (18 km/h). Give explanation to support your diagram.
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question_answer55)
In
dealing with motion of projectile in air, we ignore effect of air resistance on
motion. This gives trajectory as a parabola as you have studied. What would the
trajectory look like if air resistance is included? Sketch such a trajectory
and explain why you have drawn it that way.
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question_answer56)
A
fighter plane is flying horizontally, at an altitude of 1.5 km with speed 720
km/h. At what angle of sight (w.r.t. horizontal) when the target is seen,
should the pilot drop the bomb in order to attack the target?
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question_answer57)
(a)
Earth can be thought of as a sphere of radius 6400 km. Any object (or a person)
is performing circular motion around the axis of earth due to earth's rotation
(period 1 day). What is acceleration of object on the surface of the earth (at
equator) towards its centre? What is it at latitude \[\theta \]? How does these
acceleration compare with \[g=9.8\text{ }m/{{s}^{2}}\]?
(b) Earth also
moves in circular orbit around sun once every year with on orbital radius of\[1.5\times
{{10}^{11}}m\]. What is the acceleration of earth (or any object on the surface
of the earth) towards the centre of the sun? How does this acceleration compare
with \[g=9.8\,m/{{s}^{2}}\] ?
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question_answer58)
Given below in
column I are the relations between vectors a, b and c and
in column II are the orientations of a, b and c in the XY
plane. Match the relation in column I to correct orientations in column II
Column I
|
Column II
|
(a) \[a+b=c\]
|
(i)
|
(b) \[a-b=c\]
|
(ii)
|
(c) \[b-a=c\]
|
(iii)
|
(d) \[a+b+c=0\]
|
(iv)
|
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question_answer59)
If \[|\vec{A}|=2\]
and \[|\vec{B}|=4,\]
then match
the relations in column I with the angle \[\theta \] between A and B in
column ? II.
Column I
|
Column II
|
(a) \[\vec{A}.\,\vec{B}=0\]
|
(i) \[\theta =0\]
|
(b) \[\vec{A}.\vec{B}=+8\]
|
(ii) \[\theta =\,{{90}^{o}}\]
|
(c) \[\vec{A}.\vec{B}=4\]
|
(iii) \[\theta =\,{{180}^{o}}\]
|
(d) \[\vec{A}.\vec{B}=-8\]
|
(iv) \[\theta =\,{{60}^{o}}\]
|
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question_answer60)
If \[|A|=2\]
and \[|B|=4,\] then
match the relations in column I with the angle between A and B in column II
Column I
|
Column II
|
(a) \[|\vec{A}\times \vec{B}|=0\]
|
(i) \[\theta ={{30}^{o}}\]
|
(b) \[|\vec{A}\times \vec{B}|=8\]
|
(ii) \[\theta ={{45}^{o}}\]
|
(c) \[|\vec{A}\times \vec{B}|=4\]
|
(iii) \[\theta ={{90}^{o}}\]
|
(d) \[|\vec{A}\times \vec{B}|=4\sqrt{2}\]
|
(iv) \[\theta ={{0}^{o}}\]
|
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question_answer61)
A
hill is 500 in high. Supplies are to be sent across die hill using a canon that
can hurl packets at a speed of 125 in/s over the hill. The canon is located at
a distance of 800 m from the foot of hill and can be moved on the ground at a
speed of 2 m/s; so that its distance from the hill can be adjusted. What is the
shortest time in which a packet can reach on the ground across the hill? Take \[g=10\text{
}m/{{s}^{2}}\].
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question_answer62)
A gun can fire
shells with maximum speed \[{{\upsilon }_{0}}\] and die maximum horizontal range
that can be achieved is \[R=\,\frac{\upsilon _{0}^{2}}{g}.\]
If a target farther
away by distance \[\Delta x\] (beyond R) has to be hit with the same gun
(Fig.), show that it could be achieved by raising die gun to a height at least \[h=\,\Delta
x\,\left[ 1+\,\frac{\Delta x}{R} \right]\]
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question_answer63)
A
particle is projected in air at an angle \[\beta \] to a surface which
it self is inclined at an angle \[\alpha \] to the horizontal (Fig.).
(a) Find all
expression of range on the plane surface (distance oil the plane from the point
of projection at which particle will hit the surface).
(b) Time of flight.
(c) \[\beta \] at which range will
be maximum.
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question_answer64)
A
particle falling vertically from a height hits a plane surface inclined to
horizontal at an angle \[\theta \] with speed \[{{\upsilon }_{0}}\] and rebounds
elastically (Fig.). Find (i) die distance along the plane where it will hit
second time.
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question_answer65)
A
girl riding a bicycle with a speed of 5 m/s towards north direction, observes
rain falling vertically down. If she increases her speed to 10 m/s, rain
appears to meet her at \[{{45}^{o}}\] to the vertical. What is the speed of the rain?
In what direction does rain fall a observed by a ground based observer?
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question_answer66)
A river is flowing
due east with a speed 3m/s. A swimmer can swim in still water at a speed of 4
m/s.
(a) If swimmer
starts swimming due north, what will be his resultant velocity (magnitude and
direction)?
(b) If he wants to
start from point A on south bank and reach opposite point B on north bank.
(a) Which direction
should he swim?
(b) What will be his
resultant speed?
(c) From two
different cases as mentioned in (a) and (b) above, in which case will reach
opposite bank in shorter time?
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question_answer67)
A
cricket fielder can throw the cricket ball with a speed \[{{\upsilon }_{0}}\].
If he throws the ball while running with speed u at an angle \[\theta \] to the
horizontal, find
(a) the effective
angle to the horizontal at which the ball is projected in air as seen by a
spectator.
(b) what will be time of
flight?
(c) what is the
distance (horizontal range) from the point of projection at which the ball will
land?
(d) find \[\theta
\] at
which he should throw the ball that would maximize the horizontal range as
found in (iii).
(e) how does \[\theta
\] for maximum range change if \[u>\,{{\upsilon
}_{0}},\,\,\,u<\,{{\upsilon }_{0}}?\]
(f) how does \[\theta
\] in
(e) compare with that for \[u=O\] (i.e., \[{{45}^{o}}\])?
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question_answer68)
Motion
in two dimensions, in a plane can be studied by expressing position, velocity
and acceleration as vectors in Cartesian co- ordinates \[A\,=\,{{A}_{x}}\hat{i}+\,{{A}_{y}}\hat{j}\]
where \[\hat{i}\]
and \[\hat{j}\]
are unit
vector along \[x\] and
\[y\] directions,
respectively and \[{{A}_{x}}\] and \[{{A}_{y}}\] are corresponding components of A (fig).
Motion can also be studied by expressing vectors in circular polar co-ordinates
at \[A=\,{{A}_{r}}\hat{r}+{{A}_{\theta }}\hat{\theta },\] where \[\hat{r}=\frac{r}{r}\,\cos
\,\theta \hat{i}\,+\,\sin \,\theta \hat{j}\] and \[\hat{\theta }=-\sin \,\theta
\hat{i}+\cos \theta \hat{j}\] are unit vectors along direction in which \['r'\] and \['\theta
'\] are increasing
(a) Express \[\hat{i}\,\,and\,\hat{j}\]
in terms of \[\hat{r}\,\,and\,\hat{\theta }.\]
(b) Show that both \[\hat{r}\,\,and\,\hat{\theta
}\]are unit vectors and are perpendicular to each other.
(c) Show that \[\frac{d}{dt}\,(\hat{r})\,=\,\omega
\hat{\theta }\] where
\[\omega =\,\frac{d\theta
}{dt}\] and \[\frac{d}{dt}\,\,(\hat{\theta })\,\,=-\,\omega r\]
(d) For a particle
moving along a spiral given by \[r=a\theta \hat{r},\] where \[a=1\](unit),
find dimensions of \['a'\] .
(e) Find velocity
and acceleration in polar vector representation for particle moving along
spiral described in (d) above.
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