Kinetic Energy and Movement (Velocity).- To obtain the relation between energy and movement, let's imagine a particle of mass m moving in a straight line with initial velocity Vi . A net constant force F is then applied parallel to its movement, over a distance d. Then, the work done over the particle is W = Fd. As F = ma (a, acceleration) and using the kinetic energy formula Vf2 = Vi2 + 2ad, where Vf is the final velocity, we get:
W = Fd = mad = m[(Vf2 - Vi2) / 2d]d
That is, W = ½mVf2 - ½mVi2
It is clear we have a difference between final and initial quantities.
The kinetic energy (translational energy) of a particle is defined by physicists as the quantity ½mv2 .
Ec = ½mv2.
W can also be written
W = Ec
That is, the net work done on an object is equal to the change in his kinetic energy. This result is known as the work-kinetic energy theorem.
Let's notice W is the net work done over the object.
Example. Starting from rest, you push your 1.000 kg car over a 5 meters distance, on an horizontal ground, applying an also horizontal 400 N force. What is the car kinetic energy change?; What is its final velocity at the end of the 5 meters displacement? Disregard any friction force.
Solution. The change in kinetic energy
must be equal to the net work done on the car,
W = Fd= (400 N)(5 m) = 2.000 J.
The final velocity is obtained from the equation
W = ½mVf2 - ½mVi2, where Vi = 0.
2.000 J = ( ½ )(1000 kg)Vf2, from where Vf = 2 m/s.
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