Core Syllabus Module

System of Particles

Systematic mastery engine. Select a core micro-topic target below or simulate custom examination constraints.

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Generate random timed arrays filtered strictly by System of Particles weightage mappings.

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Micro-Topic Strategy Matrix

Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 1991

LEVELJEE Advanced

A block $A$ of mass $2 m$ is placed on another block $B$ of mass $4 m$ which in turn is placed on a fixed table. The two blocks have a same length $4 d$ and they are placed as shown in figure. The coefficient of friction (both static and kinetic) between the block $B$ and table is $μ$ . There is no friction between the two blocks. A small object of mass $m$ moving horizontally along a line passing through the centre of mass (CM) of the block $B$ and perpendicular to its face with a speed $v$ collides elastically with the block $B$ at a height $d$ above the table. (a) What is the minimum value of $v$ (call it $v_{0}$ ) required to make the block $A$ to topple? (b) If $v = 2 v_{0}$ , find the distance (from the point $P$ in the figure) at which the mass $m$ falls on the table after collision. (Ignore the role of friction during the collision.)

JEE Advanced 1997

LEVELJEE Advanced

A cart is moving along $x$ -direction with a velocity of $4 m/s$ . A person on the cart throws a stone with a velocity of $6 m/s$ relative to himself. In the frame of reference of the cart, the stone is thrown in $y$ - $z$ plane making an angle of $3 0^{\circ}$ with vertical $z$ -axis. At the highest point of its trajectory, the stone hits an object of equal mass hung vertically from branch of a tree by means of a string of length $L$ . A completely inelastic collision occurs, in which the stone gets embedded in the object. Determine ( $g = 9.8 m/s^{2}$ ) (a) the speed of the combined mass immediately after the collision with respect to an observer on the ground. (b) the length $L$ of the string such that tension in the string becomes zero when the string becomes horizontal during the subsequent motion of the combined mass.

JEE Advanced 2019

LEVELJEE Advanced

A small particle of mass $m$ moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is $L = L_{0}$ the particle speed is $v = v_{0}$ . The piston is moved inward at a very low speed $V$ such that $V ≪ v_{0}$ , where $d L$ is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct? [JEE(Advanced) 2019]

* Multiple Correct Options

(A)

The rate at which the particle strikes the piston is

v / L

(B)

After each collision with the piston, the particle speed increases by

2 V

(C)

The particle's kinetic energy increases by a factor of 4 when the piston is moved inward from

L_{0}

\frac{1}{2} L_{0}

(D)

If the piston moves inward by

d L

, the particle speed increases by

2 v \frac{d L}{L}

JEE Main 2021, 31 Aug Shift-I

LEVELJEE Advanced

A body of mass $M$ moving at speed $v_{0}$ collides elastically with a mass $m$ at rest. After the collision, the two masses move at angles $θ_{1}$ and $θ_{2}$ with respect to the initial direction of motion of the body of mass $M$ . The largest possible value of the ratio $M / m$ , for which the angles $θ_{1}$ and $θ_{2}$ will be equal, is

JEE Main 2019, 10 April Shift-I

LEVELJEE Advanced

Two particles of masses $M$ and $2 M$ , moving as shown, with speeds of $10 m/s$ and $5 m/s$ , collide elastically at the origin. After the collision, they move along the indicated directions with speed $v_{1}$ and $v_{2}$ are nearly

(A)

6.5 m/s

and

3.2 m/s

(B)

3.2 m/s

and

6.3 m/s

(C)

3.2 m/s

and

12.6 m/s

(D)

6.5 m/s

and

6.3 m/s

JEE Main 2021, 17 March Shift-II

LEVELJEE Advanced

A rubber ball is released from a height of $5 m$ above the floor. It bounces back repeatedly, always rising to $\frac{81}{100}$ of the height through which it falls. Find the average speed of the ball. (Take, $g = 10 ms^{- 2}$ )

(A)

3.0 ms^{- 1}

(B)

3.5 ms^{- 1}

(C)

2.0 ms^{- 1}

(D)

2.5 ms^{- 1}

JEE Main 2020, 9 Jan Shift-II

LEVELJEE Advanced

A particle of mass $m$ is projected with a speed $u$ from the ground at an angle $θ = \frac{π}{3}$ w.r.t. horizontal (X-axis). When it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity $u \hat{i}$ . The horizontal distance covered by the combined mass before reaching the ground is

(A)

\frac{3 3}{8} \frac{u ^{2}}{g}

(B)

\frac{3 2}{4} \frac{u ^{2}}{g}

(C)

\frac{5}{8} \frac{u ^{2}}{g}

(D)

22 \frac{u ^{2}}{g}

JEE Main 2020, 8 Jan Shift-II

LEVELJEE Advanced

As shown in figure, when a spherical cavity (centred at $O$ ) of radius 1 is cut out of a uniform sphere of radius $R$ (centred at $C$ ), the centre of mass of remaining (shaded) part of sphere is at $G$ , i.e. on the surface of the cavity. $R$ can be determined by the equation

(A)

(R^{2} + R + 1) (2 - R) = 1

(B)

(R^{2} + R - 1) (2 - R) = 1

(C)

(R^{2} - R - 1) (2 - R) = 1

(D)

(R^{2} - R + 1) (2 - R) = 1

JEE Advanced (1983)

LEVELJEE Advanced

A block of mass $M$ with a semicircular track of radius $R$ , rests on a horizontal frictionless surface. A uniform cylinder of radius $r$ and mass $m$ is released from rest at the top point $A$ (see fig.). The cylinder slips on the semicircular frictionless track. (a) How far has the block moved when the cylinder reaches the bottom (point $B$ ) of the track ? (b) How fast is the block moving when the cylinder reaches the bottom of the track?

JEE Advanced 2013

LEVELJEE Advanced

A particle of mass $m$ is projected from the ground with an initial speed $u_{0}$ at an angle $α$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed $u_{0}$ . The angle that the composite system makes with the horizontal immediately after the collision is

(A)

\frac{π}{4}

(B)

\frac{π}{4} + α

(C)

\frac{π}{4} - α

(D)

\frac{π}{2}

JEE Advanced 1987

LEVELJEE Advanced

A simple pendulum is suspended from a peg on a vertical wall. The pendulum is pulled away from the wall to a horizontal position (see fig.) and released. The ball hits the wall, the coefficient of restitution being $\frac{2}{5}$ . What is the minimum number of collisions after which the amplitude of oscillations becomes less than $6 0^{\circ}$ ?

JEE Advanced 2017

LEVELJEE Advanced

A block of mass $M$ has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at $x = 0$ , in a coordinate system fixed to the table. A point mass $m$ is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is $x$ and the velocity is $v$ . At that instant, which of the following option is/are correct?

* Multiple Correct Options

(A)

The velocity of the point mass

m

v = \frac{2 g R}{1 + \frac{m}{M}}

(B)

The

x

component of displacement of the centre of mass of the block

M

- \frac{m R}{M + m}

(C)

The position of the point mass is

x = - 2 \frac{m R}{M + m}

(D)

The velocity of the block

M

V = - \frac{m}{M} 2 g R

JEE Advanced (1997)

LEVELJEE Advanced

A cart is moving along $x$ -direction with a velocity of $4 m/s$ . A person on the cart throws a stone with a velocity of $6 m/s$ relative to himself. In the frame of reference of the cart, the stone is thrown in $y$ - $z$ plane making an angle of $3 0^{\circ}$ with vertical $z$ -axis. At the highest point of its trajectory, the stone hits an object of equal mass hung vertically from branch of a tree by means of a string of length $L$ . A completely inelastic collision occurs, in which the stone gets embedded in the object. Determine ( $g = 9.8 m/s^{2}$ ) (a) the speed of the combined mass immediately after the collision with respect to an observer on the ground. (b) the length $L$ of the string such that tension in the string becomes zero when the string becomes horizontal during the subsequent motion of the combined mass.

JEE Advanced (1984)

LEVELJEE Advanced

Two bodies $A$ and $B$ of masses $m$ and $2 m$ respectively are placed on a smooth floor. They are connected by a spring. A third body $C$ of mass $m$ moves with velocity $v_{0}$ along the line joining $A$ and $B$ and collides elastically with $A$ as shown in figure. At a certain instant of time $t_{0}$ after collision, it is found that the instantaneous velocities of $A$ and $B$ are the same. Further at this instant the compression of the spring is found to be $x_{0}$ . Determine (a) the common velocity of $A$ and $B$ at time $t_{0}$ and (b) the spring constant.

JEE Advanced 1990

LEVELJEE Advanced

An object of mass $5 kg$ is projected with a velocity of $20 m/s$ at an angle of $6 0^{\circ}$ to the horizontal. At the highest point of its path, the projectile explodes and breaks up into two fragments of masses $1 kg$ and $4 kg$ . The fragments separate horizontally after the explosion. The explosion releases internal energy such that the kinetic energy of the system at the highest point is doubled. Calculate the separation between the two fragments when they reach the ground.

JEE Advanced 1998

LEVELJEE Advanced

A wedge of mass $m$ and triangular cross-section ( $A B = B C = C A = 2 R$ ) is moving with a constant velocity ( $- v \hat{i}$ ) towards a sphere of radius $R$ fixed on a smooth horizontal table as shown in the figure. The wedge makes an elastic collision with the fixed sphere and returns along the same path without any rotation. Neglect all friction and suppose that the wedge remains in contact with the sphere for a very short time $Δ t$ during which the sphere exerts a constant force $F$ on the wedge. (a) Find the force $F$ and also the normal force $N$ exerted by the table on the wedge during the time $Δ t$ . (b) Let $h$ denote the perpendicular distance between the centre of mass of the wedge and the line of action of $F$ . Find the magnitude of the torque due to the normal force $N$ about the centre of the wedge during the interval $Δ t$ .

JEE Advanced 1999

LEVELJEE Advanced

Two blocks of masses $2 kg$ and $M$ are at rest on an inclined plane and are separated by a distance of $6.0 m$ as shown. The coefficient of friction between each block and the inclined plane is $0.25$ . The $2 kg$ block is given a velocity of $10.0 m/s$ up the inclined plane. It collides with $M$ , comes back and has a velocity of $1.0 m/s$ when it reaches its initial position. The other block $M$ after the collision moves $0.5 m$ up and comes to rest. Calculate the coefficient of restitution between the blocks and the mass of the block $M$ . [Take $sin θ \approx tan θ = 0.05$ and $g = 10 m/s^{2}$ ]

JEE Advanced 1982

LEVELJEE Advanced

Particles $P$ and $Q$ of mass $20 g$ and $40 g$ respectively are simultaneously projected from points $A$ and $B$ on the ground. The initial velocities of $P$ and $Q$ make $4 5^{\circ}$ and $13 5^{\circ}$ angles respectively with the horizontal $A B$ as shown in the figure. Each particle has an initial speed of $49 m/s$ . The separation $A B$ is $245 m$ . Both particles travel in the same vertical plane and undergo a collision. After the collision, $P$ retraces its path. (a) Determine the position $Q$ where it hits the ground. (b) How much time after the collision does the particle $Q$ take to reach the ground? (Take $g = 9.8 m/s^{2}$ ).

JEE Advanced 2025

LEVELJEE Advanced

In a scattering experiment, a particle of mass $2 m$ collides with another particle of mass $m$ , which is initially at rest. Assuming the collision to be perfectly elastic, the maximum angular deviation $θ$ of the heavier particle, as shown in the figure, in radians is:

(A)

π

(B)

tan^{- 1} (\frac{1}{2})

(C)

\frac{π}{3}

(D)

\frac{π}{6}

JEE Advanced 2021

LEVELJEE Advanced

Comprehension Passage

A projectile is thrown from a point O on the ground at an angle

4 5^{\circ}

from the vertical and with a speed

52 m/s

. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground,

0.5 s

after the splitting. The other part, t seconds after the splitting, falls to the ground at a distance x meters from the point O. The acceleration due to gravity

g = 10 m/s^{2}

Question 1:

The value of t is ___ .

Question 2:

Micro-Topic Strategy Matrix

Conservation of Linear Momentum

Head-on Collision

Oblique Collision

Centre of Mass

Motion of Centre of Mass

Chapter Elite-20 Challenge Zone

Comprehension Passage

The value of t is ___ .

The value of x is ___ .