B M Sharma Solutions for Chapter: System of Particles and Centre of Mass, Exercise 3: CONCEPT APPLICATION EXERCISE

Does the centre of mass of a rocket in free space accelerate? Explain. Can the speed of a rocket exceed the exhaust speed of the fuel? Explain.

Discuss the possibility of conservation of linear momentum of a block moving on a rough inclined plane if $\mu =\tan \theta .$

A shell is fired from a cannon with a speed of $100 \mathrm{m}/\mathrm{s}$ at an angle $30°$ with the vertical $(y$ $-\mathrm{direction}).$ At the highest point of its trajectory, the shell explodes into two fragments of masses in the ratio $1: 2.$ The lighter fragment moves vertically upwards with an initial speed of $200 \mathrm{m}/\mathrm{s}.$ What is the speed of the heavier fragment at the time of explosion?

A mud ball at rest explodes into three fragments of masses in the ratio $1: 2: 1.$ The two equal masses move with velocities $2\hat{i}+5\hat{j}-6\hat{k}$ and $-4\hat{i}+3\hat{j}+2\hat{k}.$ Find the velocity of the third mass.

(a) A rail road car of mass $M$ is moving without friction on a straight horizontal track with a velocity $u.$ A man of mass $\mathrm{m}$ lands on it normally from a helicopter. What will be the new velocity of the car?

(b) If now the man begins to run on it with speed $v_{\text{rel }}$ with respect to car in a direction opposite to motion of the car, what will be the new velocity of the car?

Two trolleys $A$ and $B$ are free to move on a level frictionless track, and are initially stationary. A man on trolley $A$ throws a bag of mass $10 \mathrm{kg}$ with a horizontal velocity of $4 \mathrm{m}{ \mathrm{s}}^{-1}$ with respect to himself on to trolley $B$ of mass $100 \mathrm{kg}.$The combined mass of trolley $A$ (excluding bag) and the man is $140 \mathrm{kg}.$ Find the ratio of velocities of trolleys $A$ and $B,$ just after the bag lands on trolley $B.$

Two identical buggies $1$ and $2$ with one man in each move along parallel rails. When the buggies are opposite to each other, the men jump in a direction perpendicular to the direction of motion of buggies, so as to exchange their places. As a consequence, buggy $1$ stops and buggy $2$ keeps moving in the same direction with its final velocity $v.$ Find the initial velocities $v_1$ and $v_2$ of buggies. Mass of each buggy (without man) equals $M,$ mass of each man is $m;$ ignore frictional effects anywhere and the buggies are constrained to move along the rails only. 

(a) A rail road flat car of mass $M$ can roll without friction along a straight horizontal track (figure). Initially, a man of mass $\mathrm{m}$ is standing on the car which is moving to the right with speed $v_0.$ What is the change in velocity of the car if the man runs to the left so that his speed relative to the car is $v_{\text{rel }}$ just before he jumps off at the left end?
(b) If there are $n$ men each of mass $\mathrm{m}$ on the car, should they all run and jump off together or should they run and jump one by one in order to give a greater velocity to the car?