Mechanical Energy > Kinetic Energy (K)

The energy possessed by any object by virtue of its motion is called its kinetic energy. The kinetic energy of an object is a measure of the work an object can do by virtue of its motion. Kinetic energy is a scalar quantity. If an object of mass m has velocity v, its kinetic energy K, then

      `k = 1 / 2 mv^2`

         `= (p^2)/ (2m)`

where k = kinetic energy, m = mass of the object, v = velocity of the object and p = mv = momentum of the object.

For example,

• When a fast cricket ball strikes the stumps, its momentum can displace the stumps, causing them to fall.
• A striker strikes a stationary coin on a carom board. The striker transfers energy to the stationary coin, setting it in motion.
• One marble striking another in a game of marbles, the moving marble transfers part of its energy to the stationary marble, making it roll.

These examples show that a moving object carries energy, and when it collides with a stationary object, it transfers some or all of its energy, causing the stationary object to move. This energy of motion is referred to as kinetic energy.

Mathematically, kinetic energy is directly related to the work done on an object. When a force F is applied to a stationary object, causing it to move a distance $s$, the work done on the object equals the kinetic energy it gains. This relationship can be expressed as:

Kinetic Energy (K.E.) = Work done on the object

K.E. = F × s

Thus, kinetic energy is a measurable quantity that reflects the work required to set an object in motion. Its study helps us understand how energy is transferred and conserved during motion, making it a vital concept in mechanics.

Suppose a stationary object of mass m moves because of an applied force. Let $u$ be its initial velocity (here u=0). Let the applied force be F. This generates an acceleration a in the object, and, after time $t$, the velocity of the object becomes equal to $v$. The displacement during this time is s. The work done on the object, W=F×s

According to Newton’s second law of motion,  

F = ma -------- (1) Similarly, using Newton’s second equation of motion 

 s = ut + `"1"/"2"` at²

However, as initial velocity is zero, u = 0

s = 0 + `"1"/"2"` at²

s =  `"1"/"2"` at² ------(2)

W = ma × `"1"/"2"` at²  ------ using equations (1) and (2)

W = `"1"/"2"`m (at)² -------(3)

Using Newton’s first equation of motion,

v = u + at
v = 0 + at
v = at
v² = (at)²  ------(4)

W = `"1"/"2"`mv² ------- using equations (3) and (4) 

The kinetic energy gained by an object is the amount of work done on the object.
K. E. = W 

 K. E. = `"1"/"2"`mv²

The energy possessed by a body due to its state of motion is called its kinetic energy.

K = \[\frac {1}{2}\] mv²

Kinetic Energy = \[\frac {1}{2}\] mass × (velocity)²

The motion of a body in a straight line path is called translational motion.

The kinetic energy of the body due to motion in a straight line is called translational kinetic energy.

If a body rotates about an axis, the motion is called rotational motion.

The kinetic energy of the body due to rotational motion is called rotational kinetic energy or simply rotational energy.

If a body moves to and fro about its mean position, the motion is called vibrational motion.

The kinetic energy of the body due to its vibrational motion is called vibrational kinetic energy or simply vibrational energy.

Statement:

According to the work-energy theorem, the increase in kinetic energy of a moving body is equal to the work done by a force acting in the direction of the moving body.

Proof:

Let a body of mass m be moving with an initial velocity u. When a constant force F is applied to the body along its direction of motion, it produces an acceleration a, and the body's velocity increases from u to v over a distance S.

Force,

F = ma

Work done by the force,

W = F × S

From the equation of motion,

\[v^2=u^2+2aS\Rightarrow S=\frac{v^2-u^2}{2a}\]

Substituting equations (i) and (iii) into (ii):

W = \[ma\times\frac{v^2-u^2}{2a}=\frac{1}{2}m(v^2-u^2)\]

Now,
Initial kinetic energy, K_i = \[\frac {1}{2}\]mu²
Final kinetic energy, K_f = \[\frac {1}{2}\]mv²

Therefore,

W = K_f − K_i

Conclusion:

Work done on the body = Increase in its kinetic energy.
Hence, the work-energy theorem is proved.