Answers to Problems on (Center Of Mass, Linear Momentum, Collision) HC Verma's Questions for Short Answer (18-25)

 Q#18

Suppose we define a quantity 'Linear Pomentum' as
Linear Pomentum = mass x speed.
The linear pomentum of a system of particles is the sum of linear pomenta of the individual particles. Can we state a principle of conservation of linear pomentum as "linear pomentum of a system remains constant if no external force acts on it"?          

Answer:
Let us consider a system of two particles of equal mass with equal speed but in opposite directions on a straight line. The center of mass of the system will remain stationary. So, the speed of the center of mass = 0.
Hence linear pomentum of the CoM = M x V = 0.
But from the definition linear pomentum of the system = mv + mv = 2mv
Hence we can not state such principle.
(Since speed v is scaler, we consider only the magnitude).

Q#19
Use the definition of linear pomentum from the previous question. Can we state the principle of conservation of linear pomentum for a single particle?          

Answer:
If no external force is applied on the particle, the speed will remain the same. Hence we can state the conservation of linear pomentum for a single particle.

Q#20
To accelerate a car we ignite petrol in the engine of the car. Since only an external force can accelerate the center of mass, is it proper to say that "the force generated by the engine accelerates the car"?          

Answer:
Technically it is not proper to say that the force generated by the engine accelerates the car because the engine does not generate a force along the acceleration. Even if it did so internal reaction force would have balanced it.

Since the movement of the piston of the engine makes a rotational motion in the wheels which in turn is used to push back the road and due to frictional force (External force) by the road, the car accelerates, hence in common language, such statement is made.

Q#21
A ball is moved on a horizontal table with some velocity. The ball stops after moving some distance. Which external force is responsible for the change in the momentum of the ball?          

Answer:
Here frictional force is the external force which is responsible for the change in the momentum of the ball.

Q#22
Consider the situation in the previous problem. Take "the table plus the ball" as the system. Friction between the table and the ball is then internal force. As the ball slows down the momentum of the system decreases. Which external force is responsible for the change in the momentum?          

Answer:
Here the external force responsible for the change in the momentum of the system is the frictional force by the ground on the table.

Q#23
When a nucleus at rest emits a beta particle, it is found that the velocities of the recoiling nucleus and the beta particle are not along the same straight line. How can this be possible in in view of the principle of conservation of momentum?          

Answer:
The motion of subatomic particles are better explained with quantum physics. Even if we try to explain the given situation with Newtonian physics, it suggests that in view of the principle of conservation of momentum there must be additional subatomic particle released during the beta emission. As we know there is antineutrino also released during the beta emission.

Q#24
A van is standing on a frictionless portion of a horizontal road. To start the engine the vehicle must be set in motion in the forward direction. How can the persons sitting inside the van do it without coming out and pushing from behind?

Answer:
The persons sitting inside the van can move the van forward to start the engine without coming out if they simultaneously move backward. In absence of an external force like friction force by the road, the center of mass of van and persons together will remain stationary. So if the persons move backward simultaneously the van will move forward to keep the center of mass at the same position and the van will start.

Q#25
In one dimensional elastic collision of equal masses, the velocities are interchanged. Can velocities in a one-dimensional collision be interchanged if the masses are not equal?            

Answer:
No, because the linear momentum will not be conserved.
For example, consider the one-dimensional collision of mass m with velocity 2v and mass 2m with velocity v before the collision.

Total momentum = m(2v) + 2m(v) = 4mv

Had the velocities been interchanged after the collision, final momentum of the system would be

= m(v) + 2m(2v) = 5mv,

which is not according to the principle of conservation of the linear momentum. So, the velocities, in this case, will not be interchanged.

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