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Kinetic Energy




(5) Kinetic energy

  • 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 v1 to v2 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 KE 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 (mv2/2) arises purely because of the motion of the body
  • In equation 7 quantity
    K2=mv22/2
    is the final Kinetic energy of the body and
    K1=mv12/2
    is the initial Kinetic of the body .Thus equation 7 becomes
    W=K2-K1=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
$K=\frac {1}{2}m \mathbf{v}. \mathbf{v}$

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

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