Analyze motion vs. gravitational energy and something thrown up in the air, and then falling, and of something rolling down a ramp. What happens to the total energy in the system?

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The motion of a body thrown upwards and that of a body rolling don a plane are totally different. Ask one specific question at a time.

A body thrown against gravity at a certain angle with the horizontal is called a projectile. When the body is thrown vertically upwards, the angle is 90 degrees. It still follows the keinematics of projectile motion, with angle `theta` = 90˚.

In projectile motion, the total mechanical energy remains constant throughout the motion. The individual values of kinetic and potential energy varies throughout the journey, though.

Kinetic energy is given by `K.E. = 1/2m*u^2`

Potential energy is given by `P.E. = mgh` (h may be obtained as `= (u^2sin^2theta)/(2g)` )

Where m is the mass ofthe body, g is the acceleration due to gravity, u is the initial velocity of theobject, and h, the height attained.

At the point of starting, h=0 there is no P.E. the entire energy is kinetic energy `= 1/2m*u^2 `

At the maximum height, there is no K.E. as the body comes to a stanstill there, the entire energy is potential energy = mgH (where H=maximum height).

`=m*g*(u^2 sin^2theta)/(2g)`

`=1/2m*u^2*1` (sin90°=1)

`=1/2m*u^2`

= total initial energy.

In between, as the body rises vertically, its K.E decreases and P.E increases, keeping the total energy constant.

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