## (5) Kinetic energy Definition and Formula

- Kinetic energy is the energy possessed by the body by virtue of its motion

- Body moving with greater velocity would posses greater K.E(Kinetic energy) in comparison of the body moving with slower velocity

- Consider a body of mass m moving under the influence of constant Force F.From newton's second law of motion

$F=ma$

Where a is the acceleration of the body

- If due to this acceleration a,velocity of the body increases from v
_{1} to v_{2} during the displacement d then from equation of motion with constant acceleration, we have

$v_2^2 -v_1^2=2ad$
or

$a= \frac {v_2^2 -v_1^2}{2d}$
Using this acceleration in Newton's second law of motion

we have

$F=m\frac {v_2^2 -v_1^2}{2d}$

or

$ Fd=\frac {m(v_2^2 -v_1^2)}{2}$

or

$Fd=\frac {1}{2}mv_2^2 -\frac {1}{2}mv_1^2$ (7)

We know that Fd is the work done by the force F in moving body through distance d

- In equation(7),quantity on the right hand side $\frac {1}{2}mv^2$ is called the kinetic energy of the body

Thus

$Kinetic \; energy =\frac {1}{2}mv^2$

- Finally we can define Kinetic Energy of the body as one half of the product of mass of the body and the square of its Speed

- Thus we see that quantity (mv
^{2}/2) arises purely because of the motion of the body

- In equation 7 quantity

K_{2}=mv_{2}^{2}/2

is the final Kinetic energy of the body and

K_{1}=mv_{1}^{2}/2

is the initial Kinetic of the body .Thus equation 7 becomes

W=K_{2}-K_{1}=ΔK (9)

- Where ΔK is the change in KE.Hence from equation (9) ,we see that work done by a force on a body is equal to the change in kinetic energy of the body

- Kinetic energy like work is a scalar quantity

- Unit of KE is same as that of work i.e Joule

- Since velocity is a vector quantity, we can expressed Kinetic energy as
$K=\frac {1}{2}m \mathbf{v}. \mathbf{v}$

Here $\mathbf{v}. \mathbf{v}$ is the scalar dot product

**Example-1**

Calculate the Kinetic energy of the object of mass 10 kg and velocity 5 m/s?

**Solution**

$Kinetic \; energy =\frac {1}{2}mv^2$

$K = \frac {1}{2} 10 \times 5^2 = 50 J$

**Example-2**

Calculate the Kinetic energy of the object of mass 10kg and velocity v = (2**i** + 3**j**) m/s at any instant?

**Solution**

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

$K= \frac {1}{2} \times 10 \times (2^2 + 3^2) =65 J$

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