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In which of the following examples of motion can the body be considered approximately a point object:
(i) a railway carriage moving without jerks between two stations.
(ii) a monkey sitting on the top of a man cycling smoothly on a circular track.
(iii) a spinning cricket ball that turns sharply on hitting the ground, and
(iv) tumbling beaker that has slipped off the edge of a table ?
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The position-time (x-t) graphs for two
children A and B returning from their school O to their homes P and Q respectively
are shown in Fig. 3.82. Choose the correct entries in the brackets below :
(a) A /B lives closer to the
school than B/A.
(b) A/B starts from the school
earlier than B/A.
(c) A/B walks faster than B/ A.
(d) A and B reach home at the
(same/different) time.
(e) A/ B overtakes B/ A on the
road (once/twice.)
Fig. 3.82
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A woman starts from her home at 9.00
A.M. walks with a speed of
on
a straight road up to her office 2.5 km away, stays at the office up to 5 P.M.
and returns home by an auto with a speed of
. Choose
suitable scales and plot the x-t graph of her motion.
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A drunkard walking in a narrow lane takes 5 steps forward and 3 steps backward, followed again by 5 steps forward and 3 steps backward, and so on. Each step is 1 m long and requires 1 s. Plot the x-t graph of his motion. Determine graphically and otherwise how long the drunkard takes to fall in a pit 13 m away from the start.
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A jet airplane travelling at the speed
of
ejects its
products of combustion at the speed of
relative to the
jet plane. What is the speed of the latter with respect to an observer on the
ground ?
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A car moving along a straight highway
with speed of
is
brought to a stop within a distance of 200 m.
What is the retardation of the car
(assumed uniform), and how long does it take for the car to stop?
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Two trains A and B of length 400 m each
are moving on two parallel tracks with a uniform speed of
in the same
direction, with A ahead of B. The driver of B decides to overtake A and
accelerates by
. If
after 50 s, the guard of B just brushes past the driver of A, what was the
original distance between them ?
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On a two-lane road, car A is travelling
with a speed of
. Two
cars B and C approach car A in opposite directions with a speed of
each. At a
certain instant, when the distance AB is equal to AC, both being 1 km, B
decides to overtake A before C does. What minimum acceleration of car B is
required to avoid an accident?
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Two towns A and B are connected by a
regular bus service with a bus leaving in either direction every T min. A man cycling
with a speed of
in
the direction A to B notices that a bus goes past him every 18 min in the
direction of his motion, and every 6 min in the opposite direction. What is the
period T of the bus service and with what speed (assumed constant) do the buses
ply on the road ?
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A player
throws a ball upwards with an initial speed of .
(i)
What is the direction of acceleration during the upward motion of the ball ?
(ii)
What are the velocity and acceleration of the ball at the highest point of its
motion ?
(iii)
Choose the x = 0 and t = 0 to be the location and time of the ball at its
highest point, vertically downward direction to be the positive direction of X-axis,
and give the signs of position, velocity and acceleration of the ball during
its upward, and downward motion.
(iv)
To what height does the ball rise and after how long does the ball return to
the player's hands ?
(Take and neglect
air resistance).
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Read each statement below carefully and state with reasons and examples, if it is true or false. A particle in one-dimensional motion:
(a) with zero speed at an instant may have non-zero acceleration at the instant,
(b) with zero speed may have non-zero velocity,
(c) with constant speed must have zero acceleration,
(d) with positive value of acceleration must be speeding up.
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A ball is dropped from a height of 90 m on a floor. At each collision with the floor, the ball loses one-tenth of its speed.
Plot the speed-time graph of its motion between t = 0 to 12 s.
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Explain clearly, with examples, the distinction between:
(a) magnitude of displacement (sometimes called distance) over an interval of time, and the total length of path covered by a particle over the same interval;
(b) magnitude of average velocity over an interval of time, and the average speed over the same interval.
[Average speed of a particle over an interval of time is defined as the total path length divided by the time interval].
(c) Show in both (a) and (b) that the second quantity is either greater than or equal to the first. When is the equality sign true ? [For simplicity, consider one-dimensional motion only].
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A man walks on a straight road from his
home to a market 2.5 km away with a speed of
. Finding the market
closed, he instantly turns and walks back home with a speed of
. What is the
(a) magnitude of average
velocity, and
(b) average speed of the man
over the interval of time
(i) 0 to 30 min,
(ii) 0 to 50 min,
(iii) 0 to 40 mm ?
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The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why?
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Fig. 3.88 shows the x-t plot of one-dimensional motion of a particle. Is it correct to say from the graph that the particle moves in a straight line for t < 0 and on a parabolic path for t > 0 ? If not, suggest a suitable physical context for this graph.
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A police van moving on a highway with a speed of
fires a bullet at a thief s car speeding away in the same direction with a speed of
. If the muzzle speed of the bullet is
, with what speed does the bullet hit the thief s car ?
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Suggest a suitable physical situation for each of the following graphs [Fig. 3.89]:
(a)
(b)
(c)
Fig. 3.89
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Figure 3.90 gives the x-t plot of a
particle executing le-dimensional simple harmonic motion. Give the signs of position,
velocity and acceleration variables of the particle at
t = 0.35, 1.25, -1.2s.
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Figure 3.91 gives the x-t plot of a
particle in one-dimensional motion. Three different equal intervals of time are
shown. In which interval is the average speed greatest, and in which is it the
least ? Give the sign of average velocity for each interval.
Fig. 3.91
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Figure 3.92 gives a speed-time graph of
a particle in motion along a constant direction. Three equal intervals of time are
shown, (a) In which interval is the average acceleration greatest in magnitude ?
(b) In which interval is the average speed greatest ? (c) Choosing the positive
direction as the constant direction of motion, give the signs of v and a in the
three intervals, (d) What are the accelerations at the points A, B, C and D ?
Fig. 3.92
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A three-wheeler starts from rest,
accelerates uniformly with
on
a straight road for 10 5, and then moves with uniform velocity. Plot the
distance covered by the vehicle during the nth second (n = 1, 2, 3...) versus
n. What do you expect this plot to be during accelerated motion: a straight
line or a parabola ?
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A boy standing on a stationary lift
(open from above) throws a ball upwards with the maximum initial speed he can,
equal to
. (i) How much
time does the ball take to return to his hands ? (ii) If the lift starts moving
up with a uniform speed of
,
and the boy again throws the ball up with the maximum speed he can, how long
does the ball take to return to his hands ?
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On a long horizontally moving belt, a child runs to and fro with a speed
(with respect to the belt) between his father and mother located 50 m apart on the moving belt.
The belt moves with a speed
. For an observer on a stationary platform outside, what is the
(i) speed of the child running in the direction of motion of the belt,
(ii) speed of the child running opposite to the direction of motion of the belt, and
(iii) time taken by the child in (i) and (ii) ?
Which of the answers alter if motion is viewed by one of the parents ?
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Two stones are thrown up simultaneously from the edge of a cliff 200 m high with initial speeds of
and
. Verify that the following graph correctly represents the time variation of the relative position of the second stone with respect to the first. Neglect air resistance and assume that the stones do not rebound after hitting the ground. Take
. Give the equations for the linear and curved parts of the plot.
Fig. 3.95
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The speed-time graph of a particle moving along a fixed direction is shown in Fig. 3.96. Obtain the distance travelled by the particle between (i) t = 0 to 10 s (ii) t = 2 to 6 s. What is the average speed of the particle in intervals in (i) and (ii) ?
Fig. 3.96
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The velocity-time graph of particle in one-dimensional motion is shown in Fig. 3.97.
Fig. 3.97
Which of the following formulae are correct for describing the motion of the particle over the time interval
to
:
(a)
(b)
(c)
(d)
(e)
.
(f)
area under the v-t curve bounded by the t-axis and the dotted line shown.
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question_answer29)
Among
the four graphs (Fig.), there is only one graph for which average velocity over
the time interval (0, T) can vanish for a suitably chosen T, Which one is it?
(a)
(b)
(c)
(d)
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question_answer30)
A
lift is coming from 8th floor and is just about to reach 4th
floor. Taking ground floor as origin and positive direction upwords for all
quantities, which one of the following is correct?
(a)
\[x<0,\,\,\upsilon ~<0,\text{ }a>0\]
(b)
\[x>0,\,\,\upsilon ~<0,\text{ }a<0\]
(c)
\[x>0,\,\,\upsilon ~<0,\text{ }a>0\]
(d)
\[x>0,\,\,\upsilon ~>0,\text{ }a<0\]
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question_answer31)
In
one dimensional motion, instantaneous speed u
satisfies \[0\le \upsilon <{{\upsilon }_{0}}\].
(a) The
displacement in time T must always take non-negative values.
(b)
The displacement \[x\] in
time T satisfies \[-\,{{\upsilon }_{0}}\,T<\,x<\,{{\upsilon
}_{0}}T\].
(c)
The acceleration is always a non-negative number.
(d)
The motion has no turning points.
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question_answer32)
A
vehicle travels half the distance L with speed \[{{V}_{1}}\] and the other half
with speed \[{{V}_{2}}\] then
its average speed is
(a)
\[\frac{{{V}_{1\,}}\,+\,{{V}_{2}}}{2}\] (b) \[\frac{2{{V}_{1\,}}\,+\,{{V}_{2}}}{{{V}_{1\,}}\,+\,{{V}_{2}}}\]
(c)
\[\frac{2{{V}_{1\,}}{{V}_{2}}}{{{V}_{1\,}}\,+\,{{V}_{2}}}\] (d) \[\frac{L({{V}_{1\,}}\,+\,{{V}_{2}})}{{{V}_{1\,}}{{V}_{2}}}\]
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question_answer33)
The
displacement of a particle is given by \[x={{(t-2)}^{2}}\] where \[x\] is in metres and t
in seconds. The distance covered by the particle in first 4 seconds is
(a)
4 m (b) 8 m
(c)
12 m (d) 16 m
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question_answer34)
At
a metro station, a girl walks up a stationary escalator in time
. If she remains
stationary on the escalator, then the escalator take her up in time
.
The time
taken by her to walk up on the moving escalator will be
(a)
(b)
(c)
(d)
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question_answer35)
The
variation of quantity A with quantity B, plotted in Fig. describes the motion
of a particle in a straight line.
(a) Quantity B may
represent time.
(b)
Quantity A is velocity if motion is uniform.
(c)
Quantity A is displacement if motion is uniform.
(d)
Quantity A is velocity if motion is uniformly
accelerated.
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question_answer36)
A graph of
various
is shown in Fig.
Choose correct alternatives from below.
(a)
The particle was re4leased from rest at t = 0.
(b)
At B, the acceleration a > 0.
(c)
At C, the velocity and the acceleration vanish.
(d)
Average velocity for the motion between A and D is positive.
(e)
The speed at D exceeds that at E.
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question_answer37)
For the
one-dimensional motion, described by
(a)
for all
(b)
for all
(c)
for all
(d)
lies between 0 and
2.
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question_answer38)
A
spring with one end attached to a mass and the other to a rigid support is
stretched and released.
(a)
Magnitude of acceleration, when just released is maximum.
(b)
Magnitude of acceleration, when at equilibrium position, is maximum.
(c)
Speed is maximum when mass is at equilibrium position.
(d)
Magnitude of displacement is always maximum whenever speed is minimum.
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question_answer39)
A
ball is bouncing elastically with a speed 1 m/s between walls of a railway
compartment of size 10 m in a direction perpendicular to walls. The train is
moving at a constant velocity of 10 m/s parallel to the directions of motion of
the ball. As seen from the ground.
(a)
the direction of motion of the ball changes every 10 seconds.
(b)
speed of ball changes every 10 seconds,
(c)
average speed of ball over any 20 second interval is fixed.
(d)
the acceleration of ball is the same as from the train.
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question_answer40)
Refer
to the graphs in of Q. No. 1 Match the following.
(a)
|
(i)
|
has \[\upsilon >0\] and \[a<0\]
throughout.
|
(b)
|
(ii)
|
has \[x>0\] throughout and
has a point with \[\upsilon =0\]
|
(c)
|
(iii)
|
has a point with zero displacement for \[t>0\]
|
(d)
|
(iv)
|
has \[\upsilon <0\] and \[a>0\]
|
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question_answer41)
A
uniformly moving cricket ball is turned back by hitting it with a bat for a
very short time interval. Show the variation of its acceleration with time.
(Take acceleration in the backward direction as positive).
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question_answer42)
Give
examples of a one-dimensional motion where
(a)
the particle moving along positive x-direction comes to rest periodically and
moves forward.
(b)
the particle moving along positive moves backward.
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question_answer43)
Give example of a
motion where \[x>0,\,\upsilon <0,\,a>0\]. At a particular instant.
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question_answer44)
An
object falling through a fluid is observed to have acceleration given by \[a=g-b\upsilon
\] where
g = gravitational acceleration and b is constant. After a long time of release,
it is observed to fall with constant speed. What must be the value of constant
speed?
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question_answer45)
A
ball is dropped and its displacement vs time graph is as shown Fig.
(displacement \[x\] is
from ground and all quantities are +ve upwards).
(a)
Plot qualitatively velocity vs time graph.
(b)
Plot qualitatively acceleration vs time graph.
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question_answer46)
A
particle executes the motion described by
\[x(t)={{x}_{0}}(1-{{e}^{-\gamma
t}});\,\,t\ge 0\].
(a)
Where does the particle start and with what velocity?
(b)
Find maximum and minimum values of \[x(t),\,\upsilon (t),\,\,a\,(t)\]. Show
that \[x(t)\] and
\[a\,(t)\] increases
with time and \[\upsilon (t)\] decreases with time.
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question_answer47)
A
bird is tossing (flying to and fro) between two cars moving towards each other
on a straight road. One car has speed of 18 km/h while the other has the speed
of 27 km/h. The bird starts moving from first car towards the other and is
moving t\with the speed of 36 km. What is the total displacement of the bird?
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question_answer48)
A
man runs across the roof-top of a tall building and jumps horizontally with the
hope of landing on the roof of the next building which is of a lower height
then the first. If his speed is 9 m/s, the (horizontal) distance between the
two buildings is 10 m and the height difference is 9 m, will he be able to land
on the next building? (take \[g=10\,m/{{s}^{2}}\])
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question_answer49)
A
ball is dropped from a building of height 45 m. Simultaneously anther ball is
thrown up with a speed 40 m/s. Calculate the relative speed of the balls as a
function of time.
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question_answer50)
The velocity ?
displacement graph of a particle is shown in Fig.
(a)
Write the relation between \[\upsilon \] and \[x\]
(b)
Obtain the relation between acceleration and displacement and plot it.
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question_answer51)
It
is a common observation that rain clouds can be at about a kilometer altitude
above the ground.
(a)
If a rain drop falls from such a height freely under gravity, what will be its
speed? Also calculate in km/h. \[(g=10\,\,m/{{s}^{2}})\]
(b)
A typical rain drop is about 4mm diameter. Momentum is mass \[x\] speed in magnitude.
Estimate its momentum when it hits ground.
(c)
Estimate the time required to flatten the drop.
(d)
Rate of change of momentum is force. Estimate how much force such a drop exert
on you.
(e)
Estimate the order of magnitude force on umbrella. Typical lateral separation
between two rain drops is 5 cm.
(Assume
that umbrella is circular and has a diameter of 1 m and cloth is not pierced
thought it)
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question_answer52)
A
motor car moving at a speed of 72 km/h cannot come to a stop in less than 3.0
while for a truck this time interval is 5.0 s. On a highway the car is behind
the truck both moving at 72 km/h. The truck gives a signal that it is going to
stop at emergency.
At
what distance the car should be from the truck so that it does not bump onto
(collide with) the truck. Human response time is 0.5s.
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question_answer53)
A
monkey climbs up a slippery pole for 3 seconds and subsequently slips for 3
seconds. Its velocity at time t is given by \[\upsilon (t)=2t\,(3t);\text{
}0<t<3\] and
\[\upsilon (t)=(t3)(6-t)\] for \[3<t<6\] in m/s. It repeats this cycle till
it reaches the height of 20 in.
(a)
At what time is its velocity maximum?
(b)
At what time is its average velocity maximum?
(c)
At what time is its acceleration maximum in magnitude?
How
many cycles (counting fractions) are required to reach the top?
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question_answer54)
A
man is standing on top of a building 100 m high. He throws two balls
vertically, one at t = 0 and other after a time interval (less than 2 seconds).
The later ball is thrown at a velocity of half the first. The vertical gap
between first and second ball is + 15 m at t = 2 s. The gap is found to remain
constant. Calculate the velocity with which the balls were thrown and the exact
time interval between their throw.
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