1. SUBJECTIVE LEVEL - II
(BRUSH UP YOUR CONCEPTS)
1. Two smooth spheres A and B, of equal radius but mass m and M, are free to move on a horizontal
table. A is projected with speed u towards B which is at rest. On impact, the line joining their centres
is inclined at an angle  to the velocity of A before impact. If e is the coefficient of restitution
between the spheres, find the speed with which B begins to move. If A’s path after impact is perpen-
dicular to its path before impact, show that
2 eM m
tan
M m

 

.
M
urel
m
L
2. A man of mass m moves on a plank of mass M with a constant
velocity rel
u . with respect to the plank, as shown in figure.
(i) If the plank rests on a smooth horizontal surface, then deter-
mine its velocity with respect to ground.
(ii) If the man travels a distance L with respect to the plank, then
find the distance traveled by the plank with respect to ground.
3. An imperfectly elastic particle is projected from a point in a horizontal place with velocity u at an
angle  to the horizon. If e be the coefficient of restitution, then calculate,
(i) the time after which it ceases to rebound from the plane
(ii) its range
(iii) tangent to the angle of projection at the nth rebound.
m1
1 2 3
V1
m2
m3
4. The centres of the spheres 1, 2 and 3 lie on a single straight line.
Sphere 1 is moving with an (initial) velocity 1
v directed along this
line and hits sphere 2. Sphere 2, acquiring after collision a velocity
2
v , hits sphere 3. Both collisions are absolutely elastic . What must
be the mass of sphere 2 for the sphere 3 to acquire maximum
velocity (the masses 1
m and 3
m of spheres 1 and 3 are known)?
5. A small empty bucket of mass M is attached to a long inextensible cord of length l. The bucket is
released from rest when the cord is in a horizontal position. In its lowest position the bucket scoops
up a mass m of water, what is the height of the swing about the lowest position?
6. Two particles A and B lighter particle has mass m, are released from infinity. They move towards
each other under their mutual force of attraction. If their speeds are v and 2v respectively find the
K.E. of the system.
7. A steel ball with a mass of m = 20 g falls from a height of h1
= 1 m onto a steel plate and rebounds to
a height of h2
= 81 cm. Find: the impulse of the force received by the plate during the impact.
8. Two identical blocks A and B of mass M each are kept on each other on a smooth horizontal plane.
There exists friction between A and B. If a bullet of mass m hits the lower block with a horizontal
velocity v and gets embedded into it. Find the work done by friction between A and B.

2. h
d
d
9. A ball is dropped from a height h above the landing and bounces
down the flight of stairs. Denoting by e the coefficient of restitu-
tion, determine the value of h for which the ball will bounce the
same height above each step.
10. A wedge of mass M and angle  can move freely on a smooth horizontal plane. A smooth sphere
of mass m strikes it in a direction perpendicular to its inclined face with a speed v and rebounds. If
the coefficient of restitution is e, then find the speed of the sphere and that of the wedge after
impact.

3. SUBJECTIVE LEVEL - III
(CHECK YOUR SKILLS)
M
m
v
1. A body of mass M with a small disc m placed on it rests on a
smooth horizontal surface. The disc is set in motion in the horizontal
direction with a velocity v. To what height relative to the initial level
will the disc rise after breaking off from the body M. Friction can
be assumed to be absent.
m
M M
h
2. The inclined surfaces of two movable wedges of the same mass M
are smoothly conjugated with the horizontal plane (as shown in
figure). A washer of mass m slides down the left wedge from a
height h. To what maximum height will the washer rise along the
right wedge? Friction should be neglected.
m1
m2
r
3. A symmetric block of mass m1
with a notch of hemispherical shape
of radius r rests on a smooth horizontal surface near the wall as
shown in figure. A small washer of mass m2
slides without friction
from the initial position shown in figure. Find the maximum velocity
of the block.
v2 = 40 m/s
B
m
m
A
600
300
v1 = 30 m/s
4. The magnitude and direction of the velocities of two iden-
tical frictionless balls before they strike each other are as
shown. Assuming e = 0.90, determine the magnitude and
direction of the velocity of the each ball after the impact.

h
H
x
5. A girl of mass M holding a bag of mass m slips from the roof of a
tall building of height H and starts falling vertically as shown in
figure. When at a height h from the ground, she notices that the
ground below her is pretty hard, but there is a pond at a horizontal
distance x from the line of fall. In order to save herself she throws
the bag horizontally (with respect to herself) in the direction oppo-
site to the pond. Calculate the minimum horizontal velocity im-
parted to the bag so that the girl lands in the water. If the girl just
succeeds to avoid the hard ground, where will the bag land ?
6 A raft of mass M with a man of mass m aboard stays motionless on the surface of a lake. The man
moves a distance l relative to the raft with velocity v (t)
 and then stops. Assuming the water resis-
tance to be negligible, find:
(a) the displacement of the raft l relative to the shore;
(b) the horizontal component of the force with which the man acted on the raft during the motion.

4. 7. A particle A of mass m moving on a smooth horizontal surface collides with a stationary particle B of
mass 2m directly, situated at a distance d from a wall. The coefficeint of restitution between A and
B and between B and the wall is e = 1/4. Calculate the distance from the wall where they collide
again. Assume that the entire motion takes place along a straight line perpendicular to the wall.
8. A wedge of mass M rests on an absolutely smooth, horizontal surface. A block of mass m is placed
on the wedge, inclined at an angle  to the horizontal. All the surfaces are frictionless. Assuming
that the system was at rest at the initial moment, find the velocity of the wedge when the block slides
down the plane through a vertical height h.
A
B
u
l
9. Two particles A and B of equal mass m each are attached by a
string of length 2l and initially placed over a smooth horizontal table
in the position shown in figure. Particle B is projected across the
table with speed u perpendicular to AB as shown in figure. Find
the velocities of each particle after the string becomes taut and the
magnitude of the impulsive tension.
10. A sphere A is of mass m and another sphere B of identical size but of mass 2m, move towards each
other with velocity ĵ
2
î  and ĵ
3
î 
 respectively. They collide when their line of centres is parallel
to .
ĵ
î  If e = 1/2, find the velocities of A and B after impact.