derive the formula for the kinetic energy of an object with mass m and moving with velocity v.
Answers
EXPLANATION
KINETIC ENERGY =
The energy posses by an object due to it's motion is known as kinetic energy.
DERIVATION
Let us consider as a body with mass m which
is at rest at smooth horizontal surface.
Let Force [F] act's on the object.
Let object from rest moves from point A to
point B and covers a displacement s
The work done by force on Object is
WORK DONE = FORCE X DISPLACEMENT
W = F X S . ...... (1)
From third equation of kinematics
V^2 = U^2 + 2AS
s = v^2 - u^2 / 2a . ....... (2)
By newton second law of motion
F = MA
From equation (1) and (2)
we get,
W = m X a ( v^2 - u^2 ) / 2a
as we have assumed object is placed at rest
therefore,
U = 0
W = mv^2 / 2
The work done appears on kinetic energy of
body
Therefore,
KE = 1/2MV^2
Hence proved
Suppose a stationary object of mass moves because of an applied force. let u be its initial velocity. (here u = 0).
Let the applied force be F. this generates an acceleration a in the object, and, after time t, the velocity of the object becomes equal to v.
The displacement during this time is s. the work done on the object- W= Fs
W= F×s
According to Newton's law of motion,
F= ma........(1)
Similarly, using Newton's second equation of motion
s= ut+ ½ at²
However, as initial velocity is zero, u= 0
s= 0+½ at².
s= ½ at²..........(2)
W= ma× ½ at²..............using equations (1) & (2)
W= ½ m× (at)²............(3)
using Newton's first equation of motion
v= u+ at
v= 0+at
v= at
v²= a²t²= (at)²...........(4)
W= ½ mv².............using equations (3) & (4)
The kinetic energy gained by an object is the amount of work done on the object.
K.E= W
K.E= ½ mv².