Class 9 > Unit # 08:Energy Sources & Transfer of Energy > Kinetic Energy

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Kinetic Energy: Definition and Derivation

Q.2 Define Kinetic Energy and derive its formula.

Definition

"Energy possessed by a body by virtue of its motion is referred to as 'Kinetic Energy'."

Formula

$\text{K.E} = \frac{1}{2}mv^2$

Where $m$ is the mass of the body and $v$ is the speed of the body.

Fig: A body of mass $m$ starting from rest ($v_i = 0$) accelerates to velocity $v$ over displacement $S$ under force $F$.

Derivation

Consider a body of mass $m$ that starts moving from rest. After a time interval $t$, its speed becomes $v$. If the initial velocity of the body is $v_i = 0$, final velocity $v_f = v$, and the displacement of the body is $S$, then we first find the acceleration of the body.

Using the 3rd equation of motion:

$2aS = v_f^2 - v_i^2$

Putting the above-mentioned values ($v_f = v$ and $v_i = 0$):

$2aS = v^2 - 0$

$a = \frac{v^2}{2S}$

Now, force is given by Newton's second law of motion:

$F = ma$

Putting the value of acceleration $a$ into the force equation:

$F = m\left(\frac{v^2}{2S}\right)$

As we know that work done is the product of force and displacement:

$\text{Work done} = F \cdot S$

Substituting the value of force $F$:

$\text{Work done} = \left[m\left(\frac{v^2}{2S}\right)\right] \cdot S$

$\text{Work done} = \frac{mv^2}{2} \quad \text{OR} \quad \text{Work done} = \frac{1}{2}mv^2$

Since the ability or capacity of doing work by a moving body is defined as its Kinetic Energy, therefore:

$\text{K.E} = \text{Work done}$

$\text{K.E} = \frac{1}{2}mv^2$

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