$\dfrac {1}{\sqrt {2}} = e^{-bt/m}$
$ln \sqrt {2} = \dfrac {bt}{m}$
$t = \dfrac {m}{b} \times \dfrac {1}{2} ln2$.

Forced vibrations: External force is acting on the body. 
Free vibration: Constant amplitude and no external force.
Damped vibration: Amplitude is not constant, it keeps on decreasing due to environmental factors of the system like air resistance.  
Therefore, correct option is B. 

The tendency of one object to force another adjoining or interconnected object into vibrating motion is referred to as a forced oscillation.

When a periodically repetitive and oscillatory force acts on an object, then the object is forced to oscillate with the frequency of the periodic force. Such oscillation is known as forced oscillation. When an object is displaced from its mean position and allowed to vibrate along its mean position then the object vibrates with its own natural frequency. This is known as free vibration or oscillation. And an oscillation in which the amplitude goes on decreasing with time is known as damped oscillation.

By pushing a child on a swing a driving force is applied which forces the swing and the child in the forward direction. The gravitational force acts as a restoring force and pulls back the child to the original position. These two forces together set them into an oscillatory motion. By pushing the child the amplitude, that is the maximum displacement, increases.

The orbital motion of the earth around the sun is not an oscillatory motion as it is not a two and fro motion about a mean position. But it is a periodic motion.

Forced vibration is an action in which periodic repetitive and oscillatory force is applied on an object. In this condition, the object is forced to vibrate with the frequency of the applied force. Hence all are forced vibrations.

A motion of an unbalanced rotating component is an example of forced vibration. As to rotate it with the unbalanced condition some external agent is required.

Resonance is an example of forced oscillation.

In vibrating motion increase in amplitude results in increase in loudness.

The oscillation of a body or system with its own natural frequency and under no external influence other than the impulse that initiated the motion is known as free oscillation.

Lesser the damping force more sharp is the resonance peak.

At resonance the amplitude of forced oscillations is maximum.

$E = \frac{1}{2}m{\omega^2}{A^2}$

In watchmakers' shop watch is oscillated with forced vibrations, so it has keeps correct time there. By itself, watch performs oscillations at natural frequency which is faster.

A wire stretched between two fixed supports, is plucked exactly in the middle and then released. It executes free vibrations.
Free vibrations are oscillations where the total energy stays the same over time. This means that the amplitude of the vibration stays the same. This is a theoretical idea because in real systems the energy is dissipated to the surroundings over time and the amplitude decays away to zero. This dissipation of energy is called damping.

The maximum speed of the vibrating particle is when particle is on mean position.
In general total energy of the system remains constant. At the mean position potential energy is minimum this implies that kinetic energy will be maximum. Hence speed will be maximum. 

Forced vibrations: External force is acting on the body. 
Free vibration: Constant amplitude and no external force.
Damped vibration: Amplitude is not constant, it keeps on decreasing due to environmental factors of the system like air resistance.  
Therefore, correct option is A. 

The acceleration due to gravity of earth at a high 'h' 

We know $ v = n \times \lambda  $

The oscillation of a pendulum about a vertical equilibrium position is an example of free vibration. As there no exciting force is present.

The correct statements are:

The force oscillation is (1) a simple harmonic motion driven by an external agency 

Statement C is true, consider the case of a guitar string mounted to the sound box. The fact that the sound box is greater than the surface area of the string means more surrounding particles will be forced into vibration causes an increase in amplitude and loudness.

The oscillations about a system of equilibrium position that occur in the absence of an external excitation are free vibrations or natural vibrations.

A person drives the paddle ball by moving his finger up and down at a accelerating frequency. This is the example of force vibration as the motion of finger force to move paddle balls.

(i) Tuning of Radio (ii) Pipe instrument (iii) Rotating machinery.

The driven force $ F= {F} _{0} \sin \omega t$

Force vibration is a type at vibration in which a body is vibrate without any external influence and vibrates with its natural frequency. So, given statement describes a free vibrations.

The vibrations which occur when work is being done on the system is called forced vibration.

A motion of reciprocating pistons in an engine is an example of force vibration. In this case motion of the piston is governed by a pressure of a gas inside ignition of a chamber.

The motion of a vehicle suspension system just after the vehicle encounters a pothole is an example of natural vibration. When suspension system encounters a pothole it is disturbed from its equilibrium position after that an oscillatory motion generates inside it to take it back its initial position but there is no external force which maintains that oscillatory motion.

The free oscillation of a body or system is due to its own natural frequency and under no external influence. But to carry out it continuously, the external impulse is needed. The child is doing this by pressing the ground.

Obviously, in the absence of such external impulse or force, free oscillations will die soon due to other negative forces i.e. damping forces.

Therefore assertion and reason both are correct and the reason is the correct explanation of assertion.

Option A is correct.

As, we know total mechanical energy $=$ maximum potential energy