1. OBJECTIVE SOLVED
1. Hail storms are observed to strike the surface of the frozen lake at 300
with the vertical and rebound
at 600
with the vertical. Assume contact to be smooth, the coefficient of restitution is :
(a)
3
1
e  (b)
3
1
e 
(c) 3
e  (d) e = 3.
Ans. (b)
Solution: Components of velocity before and after collision parallel to the plane are equal, So
v sin 60 = u sin 30 ...(1)
Components of velocity normal to the plane are related to each other
v cos 60 =e u (cos 30) ...(2)
 cot 60 = e cot 30

30
cot
60
cot
e 

3
3
1
e 
 .
3
1
e 
2. Two particles of masses m1
and m2
in projectile motion have velocities 1
v

and 2
v

respectively at
time t = 0, They collide at time 0
t . Their velocities become 1
v

and 2
v

at time 0
t
2 while still moving
in air. The value of    
2
2
1
1
2
2
1
1 v
m
v
m
'
v
m
'
v
m






 is :
(a) 0 (b)   0
2
1 gt
m
m 
(c) 2  0
2
1 gt
m
m  (d)
2
1
  0
2
1 gt
m
m  .
Ans. (c)
Solution: Before collision, the velocities of m1
and m2
at t0
are given as
1 1 0
v v gt
  

  and 2 2 0
v v gt
  

 
At t = 2 t0
, the velocities of m1
and m2
are expresed as
1 1 0
v v 2gt
 
 

 
and 2 2 0
v v 2gt
 
 

 
so, 1 1 2 2 1 1 2 2
(m v m v ) (m v m v )
 
  
   
1 2 0
2(m m )gt
 

.
3. An electron of mass m moving with a velocity v collides head on with an atom of mass M. As a result
of the collision a certain fixed amount of energy E
Δ is stored internally in the atom. The minimum
initial velocity possessed by the electron is :

2. (a)
 
Mm
E
Δ
m
M
2 
(b)
 m
m
M
E
Δ
M
2

(c)
 
Mm
E
Δ
m
M
2 
(d) none of the above is correct.
Ans. (c)
Solution: E
Δ
v
m
M
mM
2
1 2









 
Mm
E
Δ
m
M
2
v

 .
4. A bullet of mass 0.01 kg, traveling at a speed of 500 ms-1
, strikes a block of mass 2 kg, which is
suspended by a string of length 5 m, and emerges out. The block rises by a vertical distance of 0.1 m.
The speed of the bullet after it emerges from the block is :
(a) 55 ms-1
(b) 110 ms-1
(c) 220 ms-1
(d) 440 ms-1
.
Ans. (c)
Solution: By law of Conservation of Linear momentum
mu = mv +MV ...(1)
where m = mass of bullet
M = mass of block
u = velocity of bullet before collision
v = velocity of bullet after collision
V = velocity of block after collision
By law of Conservation of Energy
Mgh =
2
MV
2
1
 V = 1
.
0
x
8
.
9
x
2
 V = 1.4 ms-1
Put in (1), we get
5 = 0.01v + 2(1.4)
1
ms
01
.
0
2
.
2
v 

 v = 220 ms-1
.
5. A 6000 kg rocket is set for vertical firing. If the exhaust speed is 1000 ms-1
, the amount of gas that
must be ejected per second to supply the thrust needed to overcome the weight of the rocket is (g =
10 ms-1
).
(a) 30 kg (b) 60 kg
(c) 75 kg (d) 90 kg.
Ans. (b)
Solution:
dt
dM
v
Thrust
g
M0 


v
g
M
dt
dM 0
 =
1
kgs
1000
10
x
6000 

1
kgs
60
1000
60000
dt
dM 

 .

3. 6. A ball of mass m moving with a certain velocity collides against a stationary ball of mass m. The two
balls stick together during collision. If E be the initial kinetic energy, then the loss of kinetic energy in
the collision is
(a) E (b)
2
E
(c)
3
E
(d)
4
E
.
Ans. (b)
Solution: From COLM
mu = (m+m) v
2
u
v 
Final
2 2
2
1 1 u mu
E 2mv 2m
2 2 4 4
  
E
mu
2
1
.
E
.
K
Initial 2


2
E
4
mu
4
mu
2
mu
.
E
.
K
Final
.
E
.
K
Initial
Loss
2
2
2






7. The position of centre of mass of a system consisting of two particles of masses m1
and m2
separated
by a distance L apart from m1
(a)
2
1
2
m
m
L
m
 (b)
2
1
1
m
m
L
m

(c)
2
1
2
m
m
L
m
 (d)
2
1
1
m
m
L
m
 .
Ans. (a)
Solution: Let centre of first body be origin and line joining them is taken as x-axis
2
1
2
2
1
1
cm
m
m
r
m
r
m
r



2
1
2
m
m
L
m
0
.
m



X
Y
L
m2
m1
=
2
1
2
m
m
L
m
 .
8. Two particles of equal mass have velocities i
4
v1


 m/s and j
4
v2


 m/s. First particle has an accel-
eration 2
1
a (i j)m/s
 
 

while the acceleration of the other particle is zero. The center of mass of the
two particles moves on a :
(a) circle (b) parabola
(c) straight line (d) ellipse.
Ans. (c)
Solution:
2
1
2
2
1
1
COM
m
m
v
m
v
m
v







4. 2
v
v 2
1




2(i j)m/s
 
 
Similarly,
2
1 2
COM
a a 1
a (i j)m/s
2 2

  
   

 COM
v

is parallel to COM
a

the path will be a straight line.
9. A 1 kg ball, moving at 12 ms-1
, collides head-on with a 2 kg ball moving in the opposite direction at 24
ms-1
.If the coefficient of restitution is
3
2
, then the energy lost in the collision is :
(a) 60 J (b) 120 J
(c) 240 J (d) 480 J.
Ans. (c)
Solution: Initial momentum = Final momentum
2
2
1
1
2
2
1
1 v
m
v
m
u
m
u
m 


(1) (12) + (2) (-24) = 1v1
+ 2v2
 36
v
2
v 2
1 

 ...(i)
Further 











1
2
1
2
u
u
v
v
e
 










12
24
v
v
3
2 1
2

36
v
v
3
2 1
2 

 24
v
v 1
2 
 ...(ii)
Add (i) and (ii)
12
v
3 2 

 1
2 ms
4
v 


 1
1 ms
28
v 


 Loss = Total Initial K.E. – Total Final K.E. 





 2
mv
2
1
.
E
.
K
Since
Total Initial K. E. =      
576
2
2
1
144
1
2
1

 J
648
576
72
E1 


Similarly, total final K.E.
2
1 1
E (1)(784) (2)(16)
2 2
 
 E2
= 392 + 16

5.  E2
= 408 J
 Loss = 648 – 408
 Loss = 240 J
10. A man of mass 100 kg. is standing on a platform of mass 200 kg. which is kept on a smooth ice
surface. If the man starts moving on the platform with a speed 30 m/sec relative to the platform then
calculate with what velocity relative to the ice the platform will recoil?
(a) 5 m/sec (b) 10 m/sec
(c) 15 m/sec (d) 20 m/sec.
Ans. (b)
Solution: Let us suppose that the platform recoils with speed V towards right, when the man moves with
speed W towards left, both velocity is taken relative to ice.
If 1
V is the speed of the man relative to the platform, then
1
V W V
  .  1
W V V
 
Since, initially both man and platform are at rest, therefore
MV = mW
where, M = mass of platform. and m = mass of the man.
or, 1
MV m(V V)
 
1
mV 100 30
V
M m (200 100)

 
 
= 10 m/sec.