Question

. Derive Equation for a kinetic energy of a body of mass m moving with a velocity of v?

Answer 1

Kinetic energy is the energy possessed by an object by virtue of its motion.

K.E=1/2 mV²

Derivation:

Let us consider an object of m which is at rest lying on a table.

Let A force F acts on the object which moves the object through a distance S.

The workdone=FxS

W=FnetXS-------(1)

Let the workdone on the object causes a change in its velocity from u to V and let a be the acceleration.

From Third equation of motion:

V²-u²=2as

s=V²-u²/2a----------(2)

By Newton's Second law:

F=ma------(3)

From equation (1), (2) and (3)

W=ma*(V²-u²/2a)\=(1/2)m(V²-u²)

As we assumed object at rest, u=0

W=(1/2)mV²

we know that the kinetic energy of a body moving with certain velocity is equal to workdone on the object to acquire that velocity from rest.

∴K.E=1/2 mV²

hope this helps

Answer 2

$\huge \bold {ANSWER}$

◼The kinetic energy possessed by a moving body is equal to the amount of work which the moving body can do before coming to rest.

Suppose a body of mass m is moving with a velocity v. Its brought to the rest by applying a constant force f.

Let a be the uniform retardation produced by the force be the force and the body travel a distance S before coming to rest.

Then retardation = mass × retardation

f=m × a......(I)

Kinetic energy of the body =work done by the regarding force f in the displacement S before stopping.

Kinetic energy =retarding force × displacement.

K = f × S...........(iI)

Now to calculate the displacement S, we have

initial velocity (u) = v

final velocity (v) =0.

Since, a is the retardation = -a

From the relation ${v}^{2} = {u}^{2} + 2 a S$

0 = ${v}^{2} - 2aS. .$

Therefore,

S = $\frac{{v}^{2}}{2a}$....(III)

Substituting the values of f and S from eqns. (I) and (III) in eqns (II), we get

Kinetic energy K = f × S.

=$m × a \times \frac{{v}^{2}}{2a}$

=$\frac{1}{2} \times m \times {v}^{2}.$

K =$\frac{1}{2} \times m \times {v}^{2}.$

Kinetic energy =$\frac{1}{2} \times mass \times {velocity }^{2}..$

Thus, kinetic energy of body depends on square of its velocity of motion.