Let A be the displacement and \(\omega \)  be the angular frequency of the simple harmonic oscillator.
\(Then,\ \ a_{ 0 }=\omega ^{ 2 }A\ and\ v_{ 0 }=\omega A\)
\(\\ on\ dividing,\ \frac { { v }_{ 0 }^{ 2 } }{ { a }_{ 0 } } =\frac { { \omega }^{ 2 }{ A }^{ 2 } }{ { \omega }^{ 2 }A } =A\ or\ A=\frac { { v }_{ 0 }^{ 2 } }{ { a }_{ 0 } } \)

This is because scent vapour molecules do not travel uninterrupted, they undergo a number of collisions and trace a zig-zag path, due to which their effective displacement per unit time is small, so spreading is at a much slower rate.

The change in gravitational potential energy of a body between two given points depends only upon the position of the given points and is independent of the path followed.is due to the fact that the gravitational force is a conservative force and work done by a conservative force depends only on the position of initial and final poits and is independent of path followed.

When a cat falls to ground from a height, it stretches its body alongwith the tail so that its moment of inertia become high. Since,\(I\omega \) is to remain constant, the value of angular speed \(\omega \) decreases and therefore the cat is able to land on the ground gently.

When a child falls on a cement floor, her body comes to rest instantly.
BUt F x \(\triangle \) t = cahnge in momentum = constant.
As time of stopping  \(\triangle \) t  decreases, therefore F increases and hence child feel more pain. When she falls on a soft muddy ground in the garden, the time of stopping increases and hence F decreases and she feels lesser pain.

When a bullet is fired from a gun with its barrel directed towards the target, it starts falling downwards on account of acceleration due to gravity.
Due to which the bullet hits below the target. Just to avoid it, the barrel of the gun is lined up little above the target, so that the bullet after travelling in parabolic path hits the distant target.

An object that is in flight after being projected is called a projectile.

Hatrick is a good friend. He was concerned with his friend Peterson. He has high degree of general awareness.

(i) The horizontal range of the projectile is given by 
R = \(\frac { { u }^{ 2 }sin2\theta }{ g } \)
If angle of projection \(\theta ={ 45 }^{ o }+\alpha \quad \)and \(R={ R }_{ 1 }\)
Then, \({ R }_{ 1 }=\frac { { u }^{ 2 }sin2({ 45 }^{ o }+\alpha ) }{ g } =\frac { { u }^{ 2 } }{ g } cos2\alpha \) 
(ii) R = \(\frac { { u }^{ 2 }sin2\theta }{ g } \)
If angle of projection 
\(\theta ={ 45 }^{ o }-\alpha \quad R={ R }_{ 2 }\)
\({ R }_{ 2 }=\frac { { u }^{ 2 }sin2({ 45 }^{ o }-\alpha ) }{ g } =\frac { { u }^{ 2 } }{ g } cos2\alpha \) 

(i) The shaded portion of the car represents the distance travelled by the car in the first four seconds

The car travels with a non-uniform speed which is accelerated in nature.
(ii) The straight line portion of the graph represents the uniform motion of the car i.e. from point A to B.