what is the change in K.E as well as in P.E if any body of mass m is thrown vertically from ground explain

When a body of mass m is thrown vertically upward from the ground, it experiences changes in both kinetic energy (K.E) and potential energy (P.E) as it moves through its trajectory. Let's break down these changes step by step.

Kinetic Energy (K.E)

Kinetic energy is the energy of motion, and it is given by the formula:

K.E = (1/2)mv²

Where m is the mass of the object and v is its velocity. When the object is thrown upwards, it starts with an initial velocity (v₀) and thus has an initial kinetic energy:

K.E_initial = (1/2)mv₀²

As the object rises

As the object ascends, it slows down due to the force of gravity acting against its motion. This means that its velocity decreases, leading to a decrease in kinetic energy. At the peak of its trajectory, the velocity becomes zero, and consequently, the kinetic energy at that point is:

K.E_peak = 0

Change in K.E

The change in kinetic energy as the object moves from the ground to its peak height can be calculated as:

ΔK.E = K.E_peak - K.E_initial = 0 - (1/2)mv₀² = -(1/2)mv₀²

This negative value indicates a loss of kinetic energy as the object rises.

Potential Energy (P.E)

Potential energy, on the other hand, is the energy stored due to an object's position in a gravitational field. The formula for gravitational potential energy is:

P.E = mgh

Where h is the height above the ground. Initially, when the object is at ground level (h = 0), its potential energy is:

P.E_initial = mg(0) = 0

As the object rises

As the object moves upward, it gains height, and thus its potential energy increases. At the peak height (h_peak), the potential energy becomes:

P.E_peak = mgh_peak

Change in P.E

The change in potential energy as the object moves from the ground to its peak height can be expressed as:

ΔP.E = P.E_peak - P.E_initial = mgh_peak - 0 = mgh_peak

This positive value indicates an increase in potential energy as the object rises.

Energy Conservation

It's important to note that the total mechanical energy of the system (the sum of kinetic and potential energy) remains constant in the absence of air resistance and other non-conservative forces. Therefore, the loss in kinetic energy is equal to the gain in potential energy:

ΔK.E + ΔP.E = 0

This relationship illustrates the principle of conservation of energy, where energy is transformed from one form to another but the total remains constant.

In summary, when a body of mass m is thrown vertically, it experiences a decrease in kinetic energy as it rises and an increase in potential energy, with the total mechanical energy remaining constant throughout the motion, assuming no external forces act on it.