Question

Prove that a body in motion has its Kinetic energy equal to 1/2 mv

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Answer 1

Correct Question :-

Prove that a body in motion has its Kinetic energy equal to 1/2 mv².

Proof :-

We know that,

Work Done = Force × Displacement.

And,

Force = Mass × Acceleration.

Let's find acceleration by Third Equation of Motion.

Here, u = 0.

➨ v² - u² = 2as.

➨ v² - (0)² = 2as.

➨ v² = 2as.

➨ a = v²/2s.

Now, Put this in formula of Force.

➨ Force = m × v²/2s.

➨ Force = mv²/2s.

Now, Calculate Work Done.

As, The kinetic energy of the block increases as a result by the amount of work, We can calculate work done to calculate kinetic energy.

➨ Work done = Fs.

➨ Work done = (mv²) × s/2s.

➨ Work done = mv²/2.

➨ Work done = 1/2 mv²

Hence, Proved.

Answer 2

Answer:

Question

Prove that a body in motion has its Kinetic energy equal to 1/2 mv

Solution

${ \underline{ \Large {\sf{kinetic \: energy}}}}$

the kinetic energy of a moving body is measured by the amount of work it can do before coming to rest.

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consider an example

Suppose a body is moving with

• Mass = m
• inital velocity =v
• inital position= A
• enters into a Medium = M ( such as air )
• movement is opposed by a constant Force = F
• velocity will gradually decrease and mass comes to rest at final position B.
• final Velocity= 0

(i) In going through the distance S against the opposing force F the body has done some work.

this work is given by :-

${ \underline{\boxed{ \large\sf{work \: = force \times displacement}}}}$

At position B it has no motion and hence no kinetic energy. This means that all the kinetic energy of the body has been used up in doing work W. So, the kinetic energy must be equal to this work W.

thus,

${ \underline{ \boxed{ \sf{kinetic \: energy = W}}}} \\or \\ { \underline{ \boxed{ \sf{kinetic \: energy = F \times s}}}}$

(ii) if a body has an initial velocity 'v' final velocity 'V', Acceleration 'a' and travels a distance 's', then according to the third equation of motion :

$\sf{ : \implies \: {V}^{2} = {v}^{2} + 2as}$

In the above example we have :-

• initial velocity of the body= v ( supposed)
• final Velocity of the body = V = 0 ( the body stops)
• accleration= -a ( retardation)
• distance travelled= s

Now, putting these value in the above equation we get-:

→0 = v²-2as

→v²= 2as

From Newton's second law of motion we have

${ \underline{ \boxed{ \sf \large{f = .a}}}}$

where

• f denotes force
• a denotes Acceleration

OR

$\sf{a = \frac{f}{m} }$

putting this value of acceleration 'a' in equation v²= 2as we get:-

$\sf{ {v}^{2} = \frac{2 \times f \times s}{m}}$

$\sf{f \times s = \frac{1}{2} m {v}^{2} }$

but we know f × s = kinetic energy.

$\sf \large{kinetic \: energy = \frac{1}{2} m {v}^{2} }$

where

• m denotes mass
• v denotes velocity of the body ( or speed of the body)

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