Question

Prove that kinetic energy of a body moving with a speed v is equal to(1/2)mv^2

Answer 1

As we know that “work done in motion is called kinetic energy”
Here, we calculate the work done for a body which in motion.
Work done = Force × displacement
First of all we calculate force
Consider a body of mass m starts from rest and it's velocity become v after travelling a distance s
We have, initial velocity = 0
Final velocity = v
Displacement = s
Now by using third equation of motion
v² = u² + 2as
v² = 0 + 2as
a=v²2s(equation−1a=v²2s(equation−1)
As we know that F = ma (by Newton second law of motion)
F=m×v²2s(usingequation−1)F=m×v²2s(usingequation−1)
F=mv²2sF=mv²2s(equation−2)(equation−2)
Also, work done = Fs
W.D=mv²2s×s(using−2)W.D=mv²2s×s(using−2)
W.D=mv²2W.D=mv²2
Then by definition “work done in motion is called kinetic energy”
W.D = K.E
K.E=12mv²

Answer 2

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

_____________________

Let initial velocity = u

Let applied force = F.

Acceleration = a

Time = t

Displacement during time = s

Final velocity = v

______________________

According to the 2nd law of motion,

F = ma -------- (i)    (Work done on an object W = F * s)

By using 2nd equation of motion,

S = ut + 1/2 at²  (But, Initial velocity is 0)

S = 0 + 1/2 at²

Therefore,

S = 1/2 at² ------------ (ii)

Adding equation (i) and (ii)

W = ma * 1/2 at²

∴ W = 1/2 m(at)² ------------- (iii)

Now, By using first equation of Motion,

V = u + at [ (U) Initial velocity is 0]

V = 0 + at

V = at

∴ V² = at² -------------- (iv)

From equation (iii) and (iv)

W = 1/2 mv²

We know that, Work done by an object is equal to it's Kinetic Energy.

i.e. K.E = W

Therefore,

                             Kinetic Energy = 1/2 mv²