Kinetic Energy Formula, Definition,Concepts, examples

(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 = (2i + 3j) m/s at any instant? Solution
$K=\frac {1}{2}m \mathbf{v}. \mathbf{v}$