DC Pandey Solutions: Motion in One Dimension- 1 | DC Pandey Solutions for JEE Physics PDF Download

 Page 1


Introductory Exercise 3.1
1. Suppose a particle is moving with
constant velocity  v (along the axis of x).
Displacement of particle in time t vt
1 1
=
Displacement of particle in time t vt
2 2
=
\ Displacement of the particle in the time
interval Dt t t ( ) = -
2 1
= - vt vt
2 1
= - v t t ( )
2 1
\  Average velocity in the time interval
Dt
v t t
t t
¢ =
-
-
( )
( )
2 1
2 1
= v
Now, as the particle is moving with
constant velocity (i.e., with constant speed 
in a given direction) its velocity and speed
at any instant will obviously be v.
Ans.  True.
2. As the stone would be free to acceleration
under earth’s gravity it acceleration will
be g.
3. A second hand takes 1 min i.e., 60s to
complete one rotation (i.e., rotation by an
angle of 2p rad).
\ Angular speed of second hand
=
2
60
p rad
s
=
p
30
 rad s
-1
Linear speed of its tip = radius ´ angular 
speed
               = ´
-
2.0 cm rad s
p
30
1
=
-
p
15
1
cms
As the tip would be moving with constant
speed.
Average speed =
-
p
15
1
cms
In 15 s the second hand would rotate
through 90° i.e., the displacement of its tip 
will be r 2.
\ Modulus of average velocity of the tip of
second hand in 15 s.
 =
r 2
15
      =
-
2 2
15
1
cms
4. (a) Yes. By changing direction of motion,
there will be change in velocity and so
acceleration.
(b) (i) No. In curved path there will always
be acceleration. (As explained in the
previous answer no. 3)
(ii) Yes. In projectile motion the path of
the particle is a curved one while
acceleration of the particle remains
constant.
(iii) Yes. In curved path the acceleration
will always be there. Even if the path
is circular with constant speed the
direction of the acceleration of the
particle would every time be changing.
3
Motion in One Dimension
Page 2


Introductory Exercise 3.1
1. Suppose a particle is moving with
constant velocity  v (along the axis of x).
Displacement of particle in time t vt
1 1
=
Displacement of particle in time t vt
2 2
=
\ Displacement of the particle in the time
interval Dt t t ( ) = -
2 1
= - vt vt
2 1
= - v t t ( )
2 1
\  Average velocity in the time interval
Dt
v t t
t t
¢ =
-
-
( )
( )
2 1
2 1
= v
Now, as the particle is moving with
constant velocity (i.e., with constant speed 
in a given direction) its velocity and speed
at any instant will obviously be v.
Ans.  True.
2. As the stone would be free to acceleration
under earth’s gravity it acceleration will
be g.
3. A second hand takes 1 min i.e., 60s to
complete one rotation (i.e., rotation by an
angle of 2p rad).
\ Angular speed of second hand
=
2
60
p rad
s
=
p
30
 rad s
-1
Linear speed of its tip = radius ´ angular 
speed
               = ´
-
2.0 cm rad s
p
30
1
=
-
p
15
1
cms
As the tip would be moving with constant
speed.
Average speed =
-
p
15
1
cms
In 15 s the second hand would rotate
through 90° i.e., the displacement of its tip 
will be r 2.
\ Modulus of average velocity of the tip of
second hand in 15 s.
 =
r 2
15
      =
-
2 2
15
1
cms
4. (a) Yes. By changing direction of motion,
there will be change in velocity and so
acceleration.
(b) (i) No. In curved path there will always
be acceleration. (As explained in the
previous answer no. 3)
(ii) Yes. In projectile motion the path of
the particle is a curved one while
acceleration of the particle remains
constant.
(iii) Yes. In curved path the acceleration
will always be there. Even if the path
is circular with constant speed the
direction of the acceleration of the
particle would every time be changing.
3
Motion in One Dimension
5. (a) Time speed =
Circumference
Speed
       =
´ 2 4
1
p cm
cm/s
 =8p =25.13  s
(b)As particle is moving with constant
speed of 1 cm/s, its average speed in
any time interval will be 1 cm/s.
     | |
/
Average velocity =
r
T
2
4
                   =
æ
è
ç
ö
ø
÷
4 2
2
r
r p
speed
  =
2 2
p
 speed
 =
2 2
p
 cms
-1
= 0.9 cms
-1
| |
/
Average acceleration =
v
T
2
4
(where v = speed)
=
4 2
2
v
r
v
p
   =
2 2
2
p
v
r
     =
2 2 1
4
2
p
( )
           =
-
0.23 cm s
2
 
6. Distance = Speed ´ time
           D v t
1 1 1
=
           D v t
2 2 2
=
Average speed =
+
+
=
+
+
D D
t t
v t v t
t t
1 2
1 2
1 1 2 2
1 2
    =
´ + ´
+
( ) ( ) 4 2 6 3
2 3
=
-
5.2ms
1
  
Introductory Exercise 3.2
1. Acceleration (due to gravity).
2. s u at a
t
= + -
1
2
 is physically correct as it
gives the displacement of the particle in t
th
second (or any time unit).
s
t
= Displacement in t seconds
- displacement in ( ) t - 1 seconds
= +
é
ë
ê
ù
û
ú
- - + -
é
ë
ê
ù
û
ú
ut at u t a t
1
2
1
1
2
1
2 2
( ) ( )
Therefore, the given equation is
dimensionally incorrect.
3. Yes. When a particle executing simple
harmonic motion returns from maximum
amplitude position to its mean position
the value of its acceleration decreases
while speed increases.
4.        v t =
3 4 /
                            (given)
       
ds
dt
t =
3 4 /
…(i)
\         s t dt =
ò
3 4 /
 =
+
+
+
t
c
3
4
1
3
4
1
or       s t c = +
4
7
7 4 /
i.e.,    s t µ
74 /
Differentiating Eq. (i) w.r.t. time t,
d s
dt
t
2
2
3
4
1
3
4
=
-
Þ a t µ
-1 4 /
5. Displacement (s) of the particle 
    s= ´ + - ( ) ( ) 40 6
1
2
106
2
         = - 240 180
         =60 m (in the upward direction)
Distance covered ( ) D by the particle 
Time to attain maximum height 
Motion in One Dimension | 15
Displacement
Velocity
st = u + at – 
1
2
a
Acceleration
Page 3


Introductory Exercise 3.1
1. Suppose a particle is moving with
constant velocity  v (along the axis of x).
Displacement of particle in time t vt
1 1
=
Displacement of particle in time t vt
2 2
=
\ Displacement of the particle in the time
interval Dt t t ( ) = -
2 1
= - vt vt
2 1
= - v t t ( )
2 1
\  Average velocity in the time interval
Dt
v t t
t t
¢ =
-
-
( )
( )
2 1
2 1
= v
Now, as the particle is moving with
constant velocity (i.e., with constant speed 
in a given direction) its velocity and speed
at any instant will obviously be v.
Ans.  True.
2. As the stone would be free to acceleration
under earth’s gravity it acceleration will
be g.
3. A second hand takes 1 min i.e., 60s to
complete one rotation (i.e., rotation by an
angle of 2p rad).
\ Angular speed of second hand
=
2
60
p rad
s
=
p
30
 rad s
-1
Linear speed of its tip = radius ´ angular 
speed
               = ´
-
2.0 cm rad s
p
30
1
=
-
p
15
1
cms
As the tip would be moving with constant
speed.
Average speed =
-
p
15
1
cms
In 15 s the second hand would rotate
through 90° i.e., the displacement of its tip 
will be r 2.
\ Modulus of average velocity of the tip of
second hand in 15 s.
 =
r 2
15
      =
-
2 2
15
1
cms
4. (a) Yes. By changing direction of motion,
there will be change in velocity and so
acceleration.
(b) (i) No. In curved path there will always
be acceleration. (As explained in the
previous answer no. 3)
(ii) Yes. In projectile motion the path of
the particle is a curved one while
acceleration of the particle remains
constant.
(iii) Yes. In curved path the acceleration
will always be there. Even if the path
is circular with constant speed the
direction of the acceleration of the
particle would every time be changing.
3
Motion in One Dimension
5. (a) Time speed =
Circumference
Speed
       =
´ 2 4
1
p cm
cm/s
 =8p =25.13  s
(b)As particle is moving with constant
speed of 1 cm/s, its average speed in
any time interval will be 1 cm/s.
     | |
/
Average velocity =
r
T
2
4
                   =
æ
è
ç
ö
ø
÷
4 2
2
r
r p
speed
  =
2 2
p
 speed
 =
2 2
p
 cms
-1
= 0.9 cms
-1
| |
/
Average acceleration =
v
T
2
4
(where v = speed)
=
4 2
2
v
r
v
p
   =
2 2
2
p
v
r
     =
2 2 1
4
2
p
( )
           =
-
0.23 cm s
2
 
6. Distance = Speed ´ time
           D v t
1 1 1
=
           D v t
2 2 2
=
Average speed =
+
+
=
+
+
D D
t t
v t v t
t t
1 2
1 2
1 1 2 2
1 2
    =
´ + ´
+
( ) ( ) 4 2 6 3
2 3
=
-
5.2ms
1
  
Introductory Exercise 3.2
1. Acceleration (due to gravity).
2. s u at a
t
= + -
1
2
 is physically correct as it
gives the displacement of the particle in t
th
second (or any time unit).
s
t
= Displacement in t seconds
- displacement in ( ) t - 1 seconds
= +
é
ë
ê
ù
û
ú
- - + -
é
ë
ê
ù
û
ú
ut at u t a t
1
2
1
1
2
1
2 2
( ) ( )
Therefore, the given equation is
dimensionally incorrect.
3. Yes. When a particle executing simple
harmonic motion returns from maximum
amplitude position to its mean position
the value of its acceleration decreases
while speed increases.
4.        v t =
3 4 /
                            (given)
       
ds
dt
t =
3 4 /
…(i)
\         s t dt =
ò
3 4 /
 =
+
+
+
t
c
3
4
1
3
4
1
or       s t c = +
4
7
7 4 /
i.e.,    s t µ
74 /
Differentiating Eq. (i) w.r.t. time t,
d s
dt
t
2
2
3
4
1
3
4
=
-
Þ a t µ
-1 4 /
5. Displacement (s) of the particle 
    s= ´ + - ( ) ( ) 40 6
1
2
106
2
         = - 240 180
         =60 m (in the upward direction)
Distance covered ( ) D by the particle 
Time to attain maximum height 
Motion in One Dimension | 15
Displacement
Velocity
st = u + at – 
1
2
a
Acceleration
= =
40
10
4 s < 6 s
It implies that particle has come back
after attaining maximum height (h) given
by
                   h
u
g
=
2
2
 
  =
´
( ) 40
2 10
2
 = 80 m
\ D = + - 80 80 60 ( )
            =100 m
6.                      v t = - 40 10
\         
dx
dt
t = - 40 10
or        dx t dt = - ( ) 40 10
or x t dt = -
ò
( ) 40 10
or         x t t c = - + 40 5
2
As at t = 0 the value of x is zero.
           c =0
\           x t t = - 40 5
2
For x to be 60 m.
    60 40 5
2
= - t t
or     t t
2
8 12 0 - + =
\    t =2 s or 6 s
7. Average velocity =
Displacement in time t
t
 
             =
+ ut at
t
1
2
2
             = + u at
1
2
8.           v v at
2 1
= +
\          at v v = -
2 1
Average velocity =
Displacement in time t
t
       =
+ v t at
t
1
2
1
2
   = + v at
1
1
2
       = +
-
v
v v
1
2 1
2
=
+ v v
1 2
2
\      Ans.  True.
9.            125 0
1
2
2
= × + t gt
Þ            t =25 s
Average velocity =
125
5
m
s
       (downwards)
= 25  m/s (downwards)
10.        v t t = + - 10 5
2
         …(i
)
\       a
dv
dt
t = = - 5 2
At       t =2 s
         a = - ´ 5 2 2
            =1 m/s
2
From Eq. (i),
        
dx
dt
t t = + - 10 5
2
\ x t t dt = + -
ò
( ) 10 5
2
or x t
t t
c = + - + 10
5
2 3
2 3
As, at        t = 0 the value of x is zero
          c =0
\         x t t
t
= + - 10
5
2 3
2
3
Thus, at     t = 3 s
   x = ´ + - ( ) ( ) 10 3
5
2
3
3
3
2
3
           = + - 30 9 22.5
            = 43.5 m
11. u i
®
= 2
^
 m/s
a i j
®
= ° + ° ( cos sin )
^ ^
2 60 2 60 m/s
2
  = + ( )
^ ^
1 3 i j m/s
2
v u a
® ® ®
= + t
  = + + 2 1 3 2 i i j
^ ^ ^
( )
  = + 4 2 3 i j
^ ^
16 | Mechanics-1
60°
2
a = 2 m/s
u = 2 m/s
x
y
Page 4


Introductory Exercise 3.1
1. Suppose a particle is moving with
constant velocity  v (along the axis of x).
Displacement of particle in time t vt
1 1
=
Displacement of particle in time t vt
2 2
=
\ Displacement of the particle in the time
interval Dt t t ( ) = -
2 1
= - vt vt
2 1
= - v t t ( )
2 1
\  Average velocity in the time interval
Dt
v t t
t t
¢ =
-
-
( )
( )
2 1
2 1
= v
Now, as the particle is moving with
constant velocity (i.e., with constant speed 
in a given direction) its velocity and speed
at any instant will obviously be v.
Ans.  True.
2. As the stone would be free to acceleration
under earth’s gravity it acceleration will
be g.
3. A second hand takes 1 min i.e., 60s to
complete one rotation (i.e., rotation by an
angle of 2p rad).
\ Angular speed of second hand
=
2
60
p rad
s
=
p
30
 rad s
-1
Linear speed of its tip = radius ´ angular 
speed
               = ´
-
2.0 cm rad s
p
30
1
=
-
p
15
1
cms
As the tip would be moving with constant
speed.
Average speed =
-
p
15
1
cms
In 15 s the second hand would rotate
through 90° i.e., the displacement of its tip 
will be r 2.
\ Modulus of average velocity of the tip of
second hand in 15 s.
 =
r 2
15
      =
-
2 2
15
1
cms
4. (a) Yes. By changing direction of motion,
there will be change in velocity and so
acceleration.
(b) (i) No. In curved path there will always
be acceleration. (As explained in the
previous answer no. 3)
(ii) Yes. In projectile motion the path of
the particle is a curved one while
acceleration of the particle remains
constant.
(iii) Yes. In curved path the acceleration
will always be there. Even if the path
is circular with constant speed the
direction of the acceleration of the
particle would every time be changing.
3
Motion in One Dimension
5. (a) Time speed =
Circumference
Speed
       =
´ 2 4
1
p cm
cm/s
 =8p =25.13  s
(b)As particle is moving with constant
speed of 1 cm/s, its average speed in
any time interval will be 1 cm/s.
     | |
/
Average velocity =
r
T
2
4
                   =
æ
è
ç
ö
ø
÷
4 2
2
r
r p
speed
  =
2 2
p
 speed
 =
2 2
p
 cms
-1
= 0.9 cms
-1
| |
/
Average acceleration =
v
T
2
4
(where v = speed)
=
4 2
2
v
r
v
p
   =
2 2
2
p
v
r
     =
2 2 1
4
2
p
( )
           =
-
0.23 cm s
2
 
6. Distance = Speed ´ time
           D v t
1 1 1
=
           D v t
2 2 2
=
Average speed =
+
+
=
+
+
D D
t t
v t v t
t t
1 2
1 2
1 1 2 2
1 2
    =
´ + ´
+
( ) ( ) 4 2 6 3
2 3
=
-
5.2ms
1
  
Introductory Exercise 3.2
1. Acceleration (due to gravity).
2. s u at a
t
= + -
1
2
 is physically correct as it
gives the displacement of the particle in t
th
second (or any time unit).
s
t
= Displacement in t seconds
- displacement in ( ) t - 1 seconds
= +
é
ë
ê
ù
û
ú
- - + -
é
ë
ê
ù
û
ú
ut at u t a t
1
2
1
1
2
1
2 2
( ) ( )
Therefore, the given equation is
dimensionally incorrect.
3. Yes. When a particle executing simple
harmonic motion returns from maximum
amplitude position to its mean position
the value of its acceleration decreases
while speed increases.
4.        v t =
3 4 /
                            (given)
       
ds
dt
t =
3 4 /
…(i)
\         s t dt =
ò
3 4 /
 =
+
+
+
t
c
3
4
1
3
4
1
or       s t c = +
4
7
7 4 /
i.e.,    s t µ
74 /
Differentiating Eq. (i) w.r.t. time t,
d s
dt
t
2
2
3
4
1
3
4
=
-
Þ a t µ
-1 4 /
5. Displacement (s) of the particle 
    s= ´ + - ( ) ( ) 40 6
1
2
106
2
         = - 240 180
         =60 m (in the upward direction)
Distance covered ( ) D by the particle 
Time to attain maximum height 
Motion in One Dimension | 15
Displacement
Velocity
st = u + at – 
1
2
a
Acceleration
= =
40
10
4 s < 6 s
It implies that particle has come back
after attaining maximum height (h) given
by
                   h
u
g
=
2
2
 
  =
´
( ) 40
2 10
2
 = 80 m
\ D = + - 80 80 60 ( )
            =100 m
6.                      v t = - 40 10
\         
dx
dt
t = - 40 10
or        dx t dt = - ( ) 40 10
or x t dt = -
ò
( ) 40 10
or         x t t c = - + 40 5
2
As at t = 0 the value of x is zero.
           c =0
\           x t t = - 40 5
2
For x to be 60 m.
    60 40 5
2
= - t t
or     t t
2
8 12 0 - + =
\    t =2 s or 6 s
7. Average velocity =
Displacement in time t
t
 
             =
+ ut at
t
1
2
2
             = + u at
1
2
8.           v v at
2 1
= +
\          at v v = -
2 1
Average velocity =
Displacement in time t
t
       =
+ v t at
t
1
2
1
2
   = + v at
1
1
2
       = +
-
v
v v
1
2 1
2
=
+ v v
1 2
2
\      Ans.  True.
9.            125 0
1
2
2
= × + t gt
Þ            t =25 s
Average velocity =
125
5
m
s
       (downwards)
= 25  m/s (downwards)
10.        v t t = + - 10 5
2
         …(i
)
\       a
dv
dt
t = = - 5 2
At       t =2 s
         a = - ´ 5 2 2
            =1 m/s
2
From Eq. (i),
        
dx
dt
t t = + - 10 5
2
\ x t t dt = + -
ò
( ) 10 5
2
or x t
t t
c = + - + 10
5
2 3
2 3
As, at        t = 0 the value of x is zero
          c =0
\         x t t
t
= + - 10
5
2 3
2
3
Thus, at     t = 3 s
   x = ´ + - ( ) ( ) 10 3
5
2
3
3
3
2
3
           = + - 30 9 22.5
            = 43.5 m
11. u i
®
= 2
^
 m/s
a i j
®
= ° + ° ( cos sin )
^ ^
2 60 2 60 m/s
2
  = + ( )
^ ^
1 3 i j m/s
2
v u a
® ® ®
= + t
  = + + 2 1 3 2 i i j
^ ^ ^
( )
  = + 4 2 3 i j
^ ^
16 | Mechanics-1
60°
2
a = 2 m/s
u = 2 m/s
x
y
      | | v
®
= + 4 12
2
 =2 7 m/s
                s u a
® ® ®
= + t t
1
2
2
                  = + + ( ) ( )
^ ^ ^
2 2
1
2
1 3 2
2
i i j
                  = + + 4 2 2 3 i i j
^ ^ ^
                  = + 6 2 3 i j
^ ^
\    | | s
®
= + 36 12
                   = 4 3 m
12. Part I    v i j
®
= + ( )
^ ^
2 2t m/s …(i)
  
dv
j
¾®
=
dt
2
^
    a j
®
=2
^
 m/s
2
From Eq. (i),
       
d
dt
t
s
i j
®
= + ( )
^ ^
2 2
\       s i j
®
= +
ò
( )
^ ^
2 2t dt
        s i j
®
= + + 2
2
t t c
^ ^
Taking initial displacement to be zero.
s
®
 (at t = 1 s) = + ( )
^ ^
2 i j m
Part II  Yes. As explained below.
v i j
®
= + 2 2
^ ^
t implies that initial velocity of
the particle is 2 i
^
 m/s
2
 and the acceleration 
is 2
$
j m/s
2
\ s
®
  (at t = 1 s) = ´ + ( ) ( )
^ ^
2 1
1
2
2 1
2
i j
= + ( )
^ ^
2 i j m
13. x t = 2 and y t =
2
\   y
x
=
æ
è
ç
ö
ø
÷
2
2
or, x y
2
4 =
(The above is the equation to trajectory)
x t = 2 
\ 
dx
dt
= 2 i.e., v i
x
®
= 2
^
y t =
2
 
\
dy
dt
t = 2 i.e., v j
y
t
®
+ 2
^
Thus,           v v v
® ® ®
= +
x y
  = + ( )
^ ^
2 2 i j t m/s
a
dv
j
®
¾®
= =
dt
2
^
 m/s
2
Introductory Exercise 3.3
1. At t t =
1
 
v = tan q
As q < ° 90 , v
t
1
 is + ive.
At t t =
2
 
v
t
2
= f tan
As f > ° 90 , v
t
2
 is - ive.
Corresponding v-t graph will be
Acceleration at t t =
1
 :
a
t
1
= tan a
As a < ° 90 , a t
1
 is + ive constant.
Acceleration at t t =
2
 
a
t
2
= tan b
As b < ° 90 , a
t
2
 is + ive constant.
2. Let the particle strike ground at time t
velocity of particle when it touches ground 
Motion in One Dimension | 17
s
f
q
0
t
1
t
2 t
v
t
1
t
2 b
t
a
b
Page 5


Introductory Exercise 3.1
1. Suppose a particle is moving with
constant velocity  v (along the axis of x).
Displacement of particle in time t vt
1 1
=
Displacement of particle in time t vt
2 2
=
\ Displacement of the particle in the time
interval Dt t t ( ) = -
2 1
= - vt vt
2 1
= - v t t ( )
2 1
\  Average velocity in the time interval
Dt
v t t
t t
¢ =
-
-
( )
( )
2 1
2 1
= v
Now, as the particle is moving with
constant velocity (i.e., with constant speed 
in a given direction) its velocity and speed
at any instant will obviously be v.
Ans.  True.
2. As the stone would be free to acceleration
under earth’s gravity it acceleration will
be g.
3. A second hand takes 1 min i.e., 60s to
complete one rotation (i.e., rotation by an
angle of 2p rad).
\ Angular speed of second hand
=
2
60
p rad
s
=
p
30
 rad s
-1
Linear speed of its tip = radius ´ angular 
speed
               = ´
-
2.0 cm rad s
p
30
1
=
-
p
15
1
cms
As the tip would be moving with constant
speed.
Average speed =
-
p
15
1
cms
In 15 s the second hand would rotate
through 90° i.e., the displacement of its tip 
will be r 2.
\ Modulus of average velocity of the tip of
second hand in 15 s.
 =
r 2
15
      =
-
2 2
15
1
cms
4. (a) Yes. By changing direction of motion,
there will be change in velocity and so
acceleration.
(b) (i) No. In curved path there will always
be acceleration. (As explained in the
previous answer no. 3)
(ii) Yes. In projectile motion the path of
the particle is a curved one while
acceleration of the particle remains
constant.
(iii) Yes. In curved path the acceleration
will always be there. Even if the path
is circular with constant speed the
direction of the acceleration of the
particle would every time be changing.
3
Motion in One Dimension
5. (a) Time speed =
Circumference
Speed
       =
´ 2 4
1
p cm
cm/s
 =8p =25.13  s
(b)As particle is moving with constant
speed of 1 cm/s, its average speed in
any time interval will be 1 cm/s.
     | |
/
Average velocity =
r
T
2
4
                   =
æ
è
ç
ö
ø
÷
4 2
2
r
r p
speed
  =
2 2
p
 speed
 =
2 2
p
 cms
-1
= 0.9 cms
-1
| |
/
Average acceleration =
v
T
2
4
(where v = speed)
=
4 2
2
v
r
v
p
   =
2 2
2
p
v
r
     =
2 2 1
4
2
p
( )
           =
-
0.23 cm s
2
 
6. Distance = Speed ´ time
           D v t
1 1 1
=
           D v t
2 2 2
=
Average speed =
+
+
=
+
+
D D
t t
v t v t
t t
1 2
1 2
1 1 2 2
1 2
    =
´ + ´
+
( ) ( ) 4 2 6 3
2 3
=
-
5.2ms
1
  
Introductory Exercise 3.2
1. Acceleration (due to gravity).
2. s u at a
t
= + -
1
2
 is physically correct as it
gives the displacement of the particle in t
th
second (or any time unit).
s
t
= Displacement in t seconds
- displacement in ( ) t - 1 seconds
= +
é
ë
ê
ù
û
ú
- - + -
é
ë
ê
ù
û
ú
ut at u t a t
1
2
1
1
2
1
2 2
( ) ( )
Therefore, the given equation is
dimensionally incorrect.
3. Yes. When a particle executing simple
harmonic motion returns from maximum
amplitude position to its mean position
the value of its acceleration decreases
while speed increases.
4.        v t =
3 4 /
                            (given)
       
ds
dt
t =
3 4 /
…(i)
\         s t dt =
ò
3 4 /
 =
+
+
+
t
c
3
4
1
3
4
1
or       s t c = +
4
7
7 4 /
i.e.,    s t µ
74 /
Differentiating Eq. (i) w.r.t. time t,
d s
dt
t
2
2
3
4
1
3
4
=
-
Þ a t µ
-1 4 /
5. Displacement (s) of the particle 
    s= ´ + - ( ) ( ) 40 6
1
2
106
2
         = - 240 180
         =60 m (in the upward direction)
Distance covered ( ) D by the particle 
Time to attain maximum height 
Motion in One Dimension | 15
Displacement
Velocity
st = u + at – 
1
2
a
Acceleration
= =
40
10
4 s < 6 s
It implies that particle has come back
after attaining maximum height (h) given
by
                   h
u
g
=
2
2
 
  =
´
( ) 40
2 10
2
 = 80 m
\ D = + - 80 80 60 ( )
            =100 m
6.                      v t = - 40 10
\         
dx
dt
t = - 40 10
or        dx t dt = - ( ) 40 10
or x t dt = -
ò
( ) 40 10
or         x t t c = - + 40 5
2
As at t = 0 the value of x is zero.
           c =0
\           x t t = - 40 5
2
For x to be 60 m.
    60 40 5
2
= - t t
or     t t
2
8 12 0 - + =
\    t =2 s or 6 s
7. Average velocity =
Displacement in time t
t
 
             =
+ ut at
t
1
2
2
             = + u at
1
2
8.           v v at
2 1
= +
\          at v v = -
2 1
Average velocity =
Displacement in time t
t
       =
+ v t at
t
1
2
1
2
   = + v at
1
1
2
       = +
-
v
v v
1
2 1
2
=
+ v v
1 2
2
\      Ans.  True.
9.            125 0
1
2
2
= × + t gt
Þ            t =25 s
Average velocity =
125
5
m
s
       (downwards)
= 25  m/s (downwards)
10.        v t t = + - 10 5
2
         …(i
)
\       a
dv
dt
t = = - 5 2
At       t =2 s
         a = - ´ 5 2 2
            =1 m/s
2
From Eq. (i),
        
dx
dt
t t = + - 10 5
2
\ x t t dt = + -
ò
( ) 10 5
2
or x t
t t
c = + - + 10
5
2 3
2 3
As, at        t = 0 the value of x is zero
          c =0
\         x t t
t
= + - 10
5
2 3
2
3
Thus, at     t = 3 s
   x = ´ + - ( ) ( ) 10 3
5
2
3
3
3
2
3
           = + - 30 9 22.5
            = 43.5 m
11. u i
®
= 2
^
 m/s
a i j
®
= ° + ° ( cos sin )
^ ^
2 60 2 60 m/s
2
  = + ( )
^ ^
1 3 i j m/s
2
v u a
® ® ®
= + t
  = + + 2 1 3 2 i i j
^ ^ ^
( )
  = + 4 2 3 i j
^ ^
16 | Mechanics-1
60°
2
a = 2 m/s
u = 2 m/s
x
y
      | | v
®
= + 4 12
2
 =2 7 m/s
                s u a
® ® ®
= + t t
1
2
2
                  = + + ( ) ( )
^ ^ ^
2 2
1
2
1 3 2
2
i i j
                  = + + 4 2 2 3 i i j
^ ^ ^
                  = + 6 2 3 i j
^ ^
\    | | s
®
= + 36 12
                   = 4 3 m
12. Part I    v i j
®
= + ( )
^ ^
2 2t m/s …(i)
  
dv
j
¾®
=
dt
2
^
    a j
®
=2
^
 m/s
2
From Eq. (i),
       
d
dt
t
s
i j
®
= + ( )
^ ^
2 2
\       s i j
®
= +
ò
( )
^ ^
2 2t dt
        s i j
®
= + + 2
2
t t c
^ ^
Taking initial displacement to be zero.
s
®
 (at t = 1 s) = + ( )
^ ^
2 i j m
Part II  Yes. As explained below.
v i j
®
= + 2 2
^ ^
t implies that initial velocity of
the particle is 2 i
^
 m/s
2
 and the acceleration 
is 2
$
j m/s
2
\ s
®
  (at t = 1 s) = ´ + ( ) ( )
^ ^
2 1
1
2
2 1
2
i j
= + ( )
^ ^
2 i j m
13. x t = 2 and y t =
2
\   y
x
=
æ
è
ç
ö
ø
÷
2
2
or, x y
2
4 =
(The above is the equation to trajectory)
x t = 2 
\ 
dx
dt
= 2 i.e., v i
x
®
= 2
^
y t =
2
 
\
dy
dt
t = 2 i.e., v j
y
t
®
+ 2
^
Thus,           v v v
® ® ®
= +
x y
  = + ( )
^ ^
2 2 i j t m/s
a
dv
j
®
¾®
= =
dt
2
^
 m/s
2
Introductory Exercise 3.3
1. At t t =
1
 
v = tan q
As q < ° 90 , v
t
1
 is + ive.
At t t =
2
 
v
t
2
= f tan
As f > ° 90 , v
t
2
 is - ive.
Corresponding v-t graph will be
Acceleration at t t =
1
 :
a
t
1
= tan a
As a < ° 90 , a t
1
 is + ive constant.
Acceleration at t t =
2
 
a
t
2
= tan b
As b < ° 90 , a
t
2
 is + ive constant.
2. Let the particle strike ground at time t
velocity of particle when it touches ground 
Motion in One Dimension | 17
s
f
q
0
t
1
t
2 t
v
t
1
t
2 b
t
a
b
would be gt. KE of particle will be 
1
2
2 2
mg t
i.e., KE µ t
2
. While going up the velocity
will get - ive but the KE will remain. KE
will reduce to zero at time 2t when the
particle reaches its initial position.
KE =
1
2
2 2
mg t =
1
2
2
2
mg
h
g
                      = mgh
3. Speed of ball (just before making first
collision with floor)
             = 2gh = ´ ´ 2 10 80
             = 40 m/s
Time taken to reach ground
             =
2h
g
 =
´ 2 80
10
 = 4 s
Speed of ball (just after first collision with
floor)
= =
40
2
20 m/s
Time to attain maximum height
t =
-
-
=
20
10
2 s
\ Time for the return journey to floor = 2 s.
Corresponding velocity-time will be
4.
h
t ( )
tan
( ) -
= =
- 2
2
2 1
q
Þ h t = - 2 2 ( )
Particle will attain its initial velocity i.e.,
net increase in velocity of the particle will
be zero when,
    area under a-t graph = 0
    
( ) ( ) ( ) 1 2 2
2
2
2
0
+ ´
+
- -
=
h t
or 3 2 0
2
- - = ( ) t
or   ( ) t - = 2 3
2
or       t - = ± 2 3
or             t = ± 2 3
Ans . At time t = + 2 3 s
      (t = - 2 3 not possible).
Introductory Exercise 3.4
1. Relative acceleration of A w.r.t. B
         a g g
AB
= + - + ( ) ( ) = 0
2. Velocity of A w.r.t. B v v
A B
= -
\ Relative displacement (i.e., distance
between A and B) would be
s v v t a t
A B AB
= - + ( )
1
2
2
18 | Mechanics-1
KE
2t time
2h
g
= t
8 4
t (s)
Speed (m/s)
8 4
t (s)
Velocity (m/s)
2
a (m/s )
2
1 2
(t–2)
t
h
q
q