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Mechanics

Nov 14, 2013 (4 years and 8 months ago)

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KINEMATICS

1.

Can the magnitude of displacement be ever larger
than the path length traversed by a particle?

2.

A particle is moving in a straight line. Is it possible
for it to maintain the motion in the same direction
while the acceleration is in the reverse direction.

3.

A particle is moving with constant acceleration. Will
it always move in a straight line?

4.

The ve
locity
-
time graph of a particle moving in a
straight line is shown. What is the (i) total distance
travelled (ii) displacement in 2 s?

Ans: 2m, 0

5.

If the distance travelled by a body in time t is given
by

x = a t + b t
2
, then what will be the

instantaneous acceleration of the body?

Ans: 2b

6.

The v
-
t graph of

a particle under rectilinear motion
is shown. What is its acceleration?

a=
-
0.05ms
-
2

7.

A particle starts

from rest and moves along a
straight line in the positive x direction. The a
-
t graph
of

the particle is shown. (i) What is the velocity after
2 s ? (ii) What is the velocity after 4 s ? (iii) Is there
any reversal motion by the particle.

Ans: (i)8m/s (ii)
zero(iii)no

8.

The x

t graph starting

from rest and moving

for a
particle in a strai
ght line is shown (fig. i).

9.

(a) Draw v

t and (b) a

t graphs assuming the
acceleration between 0

4, 4

8, 8

10, 10

12 S
to be constant.

10.

The x

t graph of

an object is straight line motion
is shown. Find the instantaneous velocity of

the
object at poin
ts A and B. What is the average
velocity?
Ans: 0.5m/s

11.

From the x

t graph, find the instantaneous
velocity at F. What is the velocity at P?

(i)
-
0.75 ms
-
1

(ii) zero

12.

The x
-

t graph of a car moving in a straight line is
shown .

The curve from 0 to 10 s is of three
degrees, from 10 to 18 s is a straight line and from
18 to 20 s it is an inverted parabola. The car comes
to rest after 20 s. Draw v
-

t and a
-

t graphs.

13.

A particle moving along a straight line has x
-

t
curve given

by x = t
3

6t
2

15t + 40 where x is
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expressed in metres and t in sec. Find (a) the time
at which the velocity is zero, (b) the position and
d
istance travelled by the particle at that time (c) the
acceleration of

the particle at that time (d) the
distanc
e travelled by the particle

from t = 4 s to t = 6
S.

Ans: t=5s corresponds to zero velocity at t= 0
x=40 distance =100m (c) a= 18ms
-
2

9d) 18 m

14.

State the three equations of

motion. On what type
of

motion are these equations valid?

Ans:
V =
-
0.
5t+5.

15.

The velocity time graph of a particle in rectilinear
motion is shown. Write the equation of motion.

16.

Two balls, of

different masses are projected
vertically upwards with the same speed. Which of
the balls, heavier or light will

pass
the point of

projection with a larger speed?

17.

A balloon is rising upwards with a velocity 4 m s
-
1

and acceleration 2.5 m s
-
2
. A parcel is dropped

from
the balloon, what is the velocity of

the parcel and
what is its acceleration?

Ans: 4ms
-
1
, 10ms
-
2

18.

A car is

travelling with a speed of 100 km/h. When
a shell is

fired in the direction of

motion of the car
with a muzzle velocity 200 km/h, the velocity of

the
car is reduced by 40 km/h. What is the velocity of

shell w.r.t. the ground?

Ans:
260 km/h.

19.

A ball is thro
wn vertically upwards and it reaches a
height h after t
1

s and after t
2

s it again

reaches the
same height. What is the height h in terms of

t
1

and
t
2
?

Ans: h= ½ g t
1

t
2

20.

A ball dropped

from height h reaches the ground in
t s. After what time it has reached

h/2?

Ans:
2
t

21.

A ball is dropped
f
rom a height h and a second ball
is thrown vertically with a speed After what time will
they meet and at what height?

Ans:
g
h
, h/2

22.

A cyclist covers half

of

a

journey between two
places with speed v
1
and remaining half with speed
v
2
. What is his average speed?

Ans:
2
1
2
1
2
v
v
v
v

23.

A particle is moving along x
-
axis with x = 2
-

5t + 6t
2

?

where x is in m and t in s. What is the initial
velocity?

Ans:
-
5m/s

24.

A 100 m long train

is moving north at a speed of 16
m s
-
1
. A bird

flying south flies over the train at a
speed of

9 m s
-
1

. What time will the bird take to
cross the train ?

Ans: 4s

25.

A ball dropped

from

height h reaches the ground in
t s. After what time it has reached

h/2 ?

Ans:
2
t
s

26.

Draw approximate x

t graphs of two objects: (i)
with equal velocities moving rectilinearly in parallel.
(ii) with unequal velocities showing the time of
meeting moving rectilinearly in parallel and starting
& with a certain
displacement from each other (iii)
with velocities in opposite directions showing the
time of meeting.

UPTO PAGE 3.16

27.

Two bodies begin free fall from rest from the top of
a tower within a gap of 1 s. How long after the

first
body begins to

fall will the
two bodies be 15 m
apart.

Ans: 2s

28.

Discuss the motion of an object under free fall.
Neglect air resistance. Take acceleration due to
gravi
ty

equal to 9.8 m
s
-
2
.

Draw the a
-

t, v
-

t and

y
-
t
graphs.

29.

The reaction time of a truck driver is 0.6 s and the
truck

is travelling at 72 km/h. When the driver sees
a child at a distance of 120 m. he applies brakes so
that a deceleration of 2 ms
-

2

given to the truck.
What is the distance travelled by the truck before
stopping ? Will he be able to save the child?

Ans:
11
2m , yes

30.

A drunkard walking in a narrow lane take 5 steps
fo
rw
ard 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.

Ans: 37 s

31.

A jet airplane travelling at the speed of 500 km h
-
1

ejects its products of combustion at the speed of
1500 km h
-
1

relative to the

jet plane. What is the
speed of

th
e latter with respect to an observer on
the ground?

Ans: 1000kmh
-
1

32.

A car moving along a straight highway with speed
of 126 km h
-
1

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 ta
ke

for
the car to stop?

Ans: 11.4s

33.

Two trains A and B of length 400 m each are
moving on two parallel tracks with a uniform speed
of72 km h
-
1

in the same direction, with A ahead of
B. The driver of B decides to overtake and
accelerates by 1 m
s
-
2

If

af
t
er

50 s, the guard of

B

just brushes past the driver of

A, what was the
original distance bet
w
een them?

Ans: 1250m

34.

Read each statement below carefully and state
with reasons and examples, fit is true or false: A
particle in one
-
dimensional motion

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( a) with
zero speed at an instant may have non
-
zero acceleration at that instant.

(b) with zero speed may have non
-
zero velocity.

( c) with constant speed must have zero
acceleration.

(d) with positive value ofacceleration must be
speeding up.

35.

A ball is dropped

fro
m 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.

36.

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 pa
rticle over an interval of

time is
defined as the total path length divided by the time
interval]. 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 on
e
-
dimensional motion only].

37.

A man walks on a straight road from his home to a
market 2.5 km away with a speed of 5 km h
-
1

.
Finding the market closed, he instantly turns and
walks back home with a speed of 7.5 km h
-
1
. What
is the (a) magnitude of average v
elocity and (b)
average speed of the man over the interval of time
(i) 0 to 30 mm. (ii) 0 to 50 mm, (iii) 0 to 40 mm ?

38.

Look at the graphs (a) to (d) (fig.) carefully and
state, with reasons, which of these cannot possibly
represent one
-
dimensional motion

of

a particle.

39.

Figure shows the x
-
t plot of one
-
dimensional
motion ofa 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.

40.

A police van moving on a highway with a speed of
30 km h
-
1

fires a bullet at a thief’s car speeding
away in the same direction with a speed of 192 km
/

h . If the muzzl
e speed of the bullet is
150 m/s
, with
what speed does the bullet hit the thief’s car ?

(Note : Obtain that speed which is relevant for
damaging the thief’s car).

Ans: 105m/s

41.

Suggest a suitable

physical

situation

for each of

the

following graphs (Fig.):

42.

Two stones are thrown
up simultaneously

from the
edge of

a 200 m high with initial speeds of 15 m
/s

and 30 m
/s

. Verfy that the graph shown in

fig.
correctly represents the time variation of the relative
position of the second stone with respect to the first.
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Neglect air resist
ance and assume that the stones
do not rebound after hitting the ground. Take g =

10m/s
2
.

Give the equations for the linear and
curved

parts of

the plot.

43.

The speed
-
time graph of a particle moving along a
fixed direction is shown in fig. Obtai
n the distance
traversed by the particle between (a) t = 0 s to 10 s,
(b) t = 2 s to 6 s. What is the average speed of

the

particle over the intervals in (a) and (b)?

44.

A

stone is thrown vertically upwards with an initial
velocity of 14 m/s. Find the maxim
um height
reached and the time of ascent. Given, g = 9.8m
s

-
2
.
Ans: 1.428 s

45.

The velocity
-
time graph for a body moving along a
straight line is as shown in Fig. Find the
displacement of the body in 8 s of its motion.

Ans:
23 m

46.

A man
walks on a straight road from his home to a
market 25 km away with a speed of 5 km/h. Finding
the market closed, he instantly turns and walks
home with a speed of 75 km/h. What is the
magnitude of the average velocity and average
speed of the man over the
interval of time (i) 0 to 30
min, (ii) 0 to 50 min and (iii) 0 to 40 min? Ans: (i)
5km/h, 5km/h (ii) 6km/h (iii)1.875
km/h
,5.625km/h
.

47.

The position
-
time graph for a dancer demonstrating
dance steps along a straight line is as shown in Fig.

(a)

What are the ave
rage speed and average
velocity for each dance step? Ans:
1m/s,0m/s, 1.33m/s, 1.33m/s, 2.75m/s,
-
2.75m/s, 1m/s,1m/s

(b)

What is the average velocity of the dancer
during the time interval between t = 4.5 to t
= 9 s? Ans:
-
0.89m/s.

48.

Fig. shows velocity
-
time gra
ph for the motion of an
object. (a) Compute the acceleration for each phase
of the motion. (b) Describe how the object moves
during the last time segment.

Ans: OA

0,AB
-

4m/s
2
, BC
-

-
4m/s
2
.
DE
-
3
m/s
2
,
EF
-

a=0, v=
-
4m/s.

49.

The velocity
-
time graph of an object moving along a
straight line is as shown in the Fig.
Calculate
the distance covered by the object: (i) between t = 0
to t = 5 s and (ii) between t = 0 to t = 10 s. Ans:
80m, 130m.

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50.

The velocity
-
time graph of a particle
moving along a
fixed direction as
shown in Fig. Obtain the distance traversed by the
particle between (a) t = 0 to 10 s, (b) t = 2 to 6 s.
What is the average speed of the particle over the
intervals in (a) and (b)? Ans. (a) 60 m ; 6 m/s (ii) 36
m ; (iii
) 9 m/s.

51.

On a foggy day, two car drivers spot each

other,
when they are just 80 m apart. They are travelling
at

72 km
/

h

and 60 km
/

h respectively. Both of them

simultaneously apply brakes, which retard both the
cars at

a rate of 5 m
/s
2

.
Determine whether

they
avert the collision

or not.

Ans: the collision will be
averted.

52.

A car starts from rest and accelerates uniformly for
10 s to a velocity of 8 m/s. It then runs at a constant
velocity and is finally brought to rest in 64 m with a
constant retardation.

The total distance covered by
the car is 584 m. Find the value of acceleration,
retardation and total time taken. Ans; 0.8m/s
2
,
-
0.5m/s
2
, 86s.

53.

A body covers 4 m in 3rd second and 12 m in 5th
second. If the motion is uniformly accelerated, how
far will it
travel in the next three seconds? Ans: 60m

54.

The velocity

time

graph
of an object moving along

a

straight line is as shown in Fig. Find (a) the
distance covered and (b) the displacement by the
object in time interval between t = 0 to t = 5 s. [Ans.
(a) 50
m

(b) 30 m
/s.

55.

The velocity
-
time graph of an object moving along a
straight line is as shown in Fig. Find the distance
covered by the object in time t = 0 to t = 7 s and the
maximum value of the acceleration during this
interval.[Ans. 170
m
, 40
m/s]

56.

A
balloon is ascending at the rate of 14 m
/
s at a
height of 98 m above the ground, when a packet is
dropped from the balloon. After how much time and
with what velocity, does it reach the ground?

57.

The displacement (in metre) of a particle, moving
along X
-
axis

is given by x = 18 t + 5 t
2
. Calculate (i)
the instantaneous velocity at t = 2 s, (ii) the average
velocity between t = 2 s and t = 3 s and (iii) the
instantaneous acceleration.(i) 38m/s (ii) 43m/s (iii)
10m/s
2
.

58.

A parachutist bails out from an aeroplane

and after
dropping through a distance of 40 m, he opens the
parachute and decelerates at 2 m
/
s
2
. If he reaches
the ground with a speed of 2 m
/
s, how long is he in
the air. At what height did he bail out from the
plane?

Ans: 111/7 sec
.

59.

A ball is allowed to

fall from the top of a tower 200
m high. At the same instant, another

ball is thrown
vertically upwards from the bottom of th
e tower with
a velocity of 40 m/
s. When and where the two balls
meet?

Ans: 77.5m

60.

Ball A is thrown upward with a speed of 35 m
/
s fr
om
ground, while at the same time another ball B is
dropped from a height of 100 m with a speed 10
m/s along the same straight

line. By choosing
positive direction of X
-
axis and origins for position
and time, find the height
from
ground,

where

the

tw
o
ball
s m
eet.

Take

g =10ms
-
2
.
Ans:53.09m .

61.

A stone let fall from the top of a tower traverses a
distance of 24•5 m in the last second of its fall. Find
the height of the tower.

Ans: 44.1 m.

62.

A body travels a distance of 2 m in 2 s and 2.8 m in
next 4s. What will b
e speed of the body at the end
of 10th second from the start. [Ans. 0.1
m/s
]

63.

Show that the distances traversed by a freely falling
body during equal intervals of time stand to one
another in the same rati3 as the odd numbers
beginning with unity i.e. 1 : 3

: 5 : 7

64.

Two bodies fall freely from different heights and
reach the ground simultaneously. The time of
descent for the first body is 2 s and for the second
is 1 s. At what height was the first body situated,
when the other began to fall?
[Ans. 14.7 m]

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65.

A b
oy on a cliff 49 m high drops a stone and one
second later throws a second stone after the first.
They

both hit the ground at the same time. How fast
did the boy throw the second stone ? Neglect air
resistance. [Ans. 1207
m/s
]

66.

A stone is dropped from the e
dge of a roof and is
observed to pass a window, vertically below i
t. The
stone takes 0.1 s to fall
between the top and the
bottom of the window. If the window is of 2
m

height, how far is the roof above the top of the
windo
w

? [Ans. 19
.4 m]

67.

A stone is thro
wn vertically upwards with an initial
velocity of 14 m
/
s. Find the maximum height
reached and the time of
ascent
. Given g
=
9
.
8
m/
s
2
.

[Ans. 10
m
, 1•429 s]

68.

When it is at a height of 98
m
, a packet is dropped
from it. What is the velocity of the packet, when
it
strikes the

ground ? [Ans. 44.1 m
/
s]

69.

From the top of a

tower 100 m in height, a ball is
dropped and at the same time another ball is
projected vertically upwards from the ground with a
velocity of25 m
/
s. Find when and where

t
he two
balls will meet ?

(g = 9•8
m/s
2

[Ans. 4 s, 78•4
m
(from the top)]

70.

Two balls are thrown simultaneously. A vertically
upwards with a speed of 20
m/s

from the ground
and B, vertically downwards from a height of

4
0

m

with the same speed and along the same line of
motion. At what points do the two balls collide ?
Take g = 9.8
m/s
2
.

[Ans. 15.1 in (above the ground)]

71.

A ship is sailing westward at 8 m
/
s. While trying to
fix a bolt at the top of the mast, the sailor drops
the
bolt. If tire mast of the ship is19•6 m high, where will
the bolt hit the deck?

Ans: At its foot.(16m)

72.

A projectile
foll
ows a parabolic path. At what point
of its path, the speed of the projectile is
minimum

?

73.

A bulb is accidentally dropped, from the
ceiling of

a
train moving with constant velocity. Wha
t
shape of

the path will be seen

frorn an observer on the
ground?

74.

At what angle be a projectile thrown with the
horizontal that its horizontal range is equal to the
maximum height attained?

75.

A projectil
e is thrown with a velocity v =
j
i
ˆ
4
ˆ
3

in
m/s. What is the velocity at the top?

76.

A body is projected at an angle of35° with the
horizontal. At what other angle can it be thrown to
achieve the same horizontal range?

77.

A ball is dropped

frorn

the top
o
f

the tower and
another ball is thrown horizontally at the same time.
Which ball will reach the ground

fir
st?

78.

A ball is dropped

frorn the top
o
f

the tower and
another ball is thrown horizontally at the same time.
Which ball will reach the ground

fi
rst?

79.

A ball is projected with a velocity I 9.6 rn at an angle
of

3
0
°. After what time will the bull mo
v
e at right
angles to its direction of mo
ti
on?

80.

There are two directions
θ

and (90°
-

θ
) for the
same range for a projectile. For which directio
n
more
time is taken by the proj
ectile?

81.

A projectile is thrown at an angle of 45° with the
horizontal. What is the horizontal range of the
particle ?

82.

A man can throw a stone upto a distance x, to what
height can he throw?

83.

Two paper screens A and B are separated
by a
distance of 100 m. A bullet pierces A and then B.
The hole in B is 0.1 m below the hole in A .

If the bullet is travelling horizontally at the time of

hitting the screen A, calculate the velocity of the

bullet, when it hits the screen A. Neglect the

resistance of paper and air and take g = 9•8 m/s
2
.

700m/s.

84.

A projectile is fired horizontally with a velocity of 98
m/s from the hill 490 m high. Find (i) time taken to
reach the ground, (ii) the distance o
f the target from
the hill and (iii) the velocity with which the body
strikes the ground.

Ans: t=10s,x=980m, 98
2
m/s.

85.

A ball rolls off the top of a stairway with a horizontal
velocity u m s
-
i. If the steps are h metres high and b
metre
s wide, show that the ball will hit the edge of
the nth step ; if

2
2
gb
hu
2
n

.

86.

A body is projected such that its kinetic energy at
the top is 3/4 th of its initial kinetic energy. What is
the initial angle of projection of the projectile with
t
he hor
iz
ontal? [Ans. 30°]

87.

A shot is fired with a velocity 100 m
/
s in a direction
making an angle 60
0

with the vertical. Calculate its
time of flight, the maximum height reached and its
horizontal range
.
Ans: 10.2s, 127.55m, 883.67m.

88.

A projectile is thrown from a point 39•2 m away
from the foot of a building 19.6, m high and just
reaches the top horizontally. Find the velocity of
projection and the angle of throw.

27.72m/s, 45
0

89.

A cricketer can throw a ball to a maximum
horizontal dis
tance of 100 m. How much high above
the ground can the cricketer throw the same ball?

Ans: 50m.

90.

What is a projectile ? Obtain the formulae (a) for
time taken to reach maximum height H (b)for
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maximum height H (c)for time of flight (d)for
horizontal range (e)for the equation of the trajectory
for
Find the angle of projection at which the
horizontal range and maximum height of a projectile
are equal. Ans: tanθ = 4.

91.

Prove that the maximum horizont
al range is four
times the maximum height attained by the
projectile, when fired at an inclination so as to have
maximum horizontal range.

92.

A man can jump on moon six times as high as on
earth. Why?

93.

An aeroplane is flying horizontally at a height of 490
m w
ith a velocity of 360 km /h. A bag containing
ration is to be dropped to the jawans on the ground.
How far from them should the

bag

be released, so
that it falls directly over
them ? [Ans. 1,000 m]

94.

Find the angle of projection so that a body when
projected

has the horizontal range equal to the
maximum height attained. [Ans. 76°]

95.

Prove that a gun will shoot three times as high,
when its angle of elevation is 60° as when it is 30°,
but cover the same horizontal range.

96.

A projectile is thrown horizontally from
the top of a
tower and strikes the ground after 3 s at an angle of
45° with the horizontal. Find the height of the tower
and speed with which the body was projected.
Given g = 9•8 m/s
2

[Ans. 444 m ; 29•4 m s]

97.

A ball rolls off the top of a stairway with
horizontal
velocity of 1.8 m/s. The steps are 0.24 m high and
0.2 m wide. Which step will the ball hit first? Given g
= 9•8 m
/s
2

fAns. Fourth step]

98.

What is the angular velocity of a flywheel making
300 r.p.m. in rad ?[Ans. 10
π

/s]

99.

ane propeller are rotating at
the rate of 600 revolutions per minute. Calculate the
/s
. [Ans. 20
π

/
s]

100.

Obtain an expression

for the centripelal
acceleration in a un(form circular motion.

101.

The angular speed of a wheel is 88 rad/s.
C
alculate the number of revolutions made by it in
one second.
[Ans. 14 r.p.s.]

102.

An insect moves in a circular groove of
radius 12 cm and completes 7 revolutions in 100 s
at a uniform pace. Calculate (i) angular speed, (ii)
magnitude of

velocity and (iii) mag
nitude of
centripetal acceleration. [Ans. (i) 044 rad ; (ii) 5•28
cm
/
s ; (iii) 2•32 cm s
-
2
]

103.

A stone tied to one end of a string 70 cm
long is whirled in horizontal circle with a constant
speed. If the stone makes 10 revolutions in 20 s,
what is the directi
on and magnitude of the
acceleration of the stone?[Ans. along radius of the
circular path, 690.87 cm/s
-
2
]

104.

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 alw
ays
along the radius of the circle towards the centre. (b)
The velocity vector ofa 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.

105.

A cyclist

is riding with a speed of 27 km/
h.
As he approaches a c
ircular