Question

kinetic energy depends on a mass of body .Using dimensional analysis derive equation for kinetic energy ?


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Answer 1

Answer:

Correction: Kinetic Energy depends both on mass and velocity of the body.

We know that Energy is denoted [ M¹ L²T⁻² ]

⇒ Kinetic Energy = k [ M ] ᵃ [ V ]ᵇ

Here, 'k' is a constant

⇒ [ V ] ᵇ = [ M⁰ L¹ T⁻¹ ] ᵇ

⇒ Kinetic Energy = k [ M ] ᵃ [ M⁰ L¹ T⁻¹ ] ᵇ

⇒ [ M¹ L²T⁻² ] = k [ M ] ᵃ [ L ] ᵇ [ T ] ⁻ᵇ

Comparing the equations on both sides we get,

⇒ M¹ = Mᵃ

⇒ a = 1

Similarly,

⇒ L² = Lᵇ

⇒  b = 2

⇒ T⁻² = T⁻ᵇ

⇒ b = 2

Hence the formula for Kinetic Energy is:

⇒ K.E. = k [ M ] ᵃ [ V ] ᵇ

⇒ K.E. = k [ M ] ¹ [ V ] ²

This is the dimensional formula of Kinetic Energy.

On experiments, we get to know that, value of k is 1/2.

Hence K.E. = 1/2 mv²

Hope it helps !!

Answer 2

Explanation:

The energy depends on a mass of body and velocity with which it is moving such that,

$k\propto [m]^a[v]^b$ ......(1)

Dimensional formula of mass is, $[m]=[M]$

Dimensional formula of velocity, $[v]=[LT^{-1}]$

So,

$k\propto [M]^a[LT^{-1}]^b\\\\k\propto [M^aL^bT^{-b}]$

The dimensional formula of energy is, $[E]=[ML^2T^{-2}]$

So,

$[ML^2T^{-2}]\propto [M^aL^bT^{-b}]$

On comparing both sides,

a = 1, b =  2

So, equation (1) becomes,

$k\propto mv^2\\\\k=\dfrac{1}{2}mv^2$

Hence, this is the required solution.

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