Question

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Answer 1

QUESTION :-

Two bodies have the kinetic energies in the ratio of 4 : 1  . are moving with equal linear momentum . The ratio of their masses are ?

ANSWER :-

Let the kinetic energy of the first body be K₁ and the kinetic energy of the second body be K₂ .

K₁ : K₂ = 4 : 1

⇒ K₁ / K₂ = 4 / 1

⇒ K₁ = 4 K₂ ----( 1 )

We know that K = 1/2 m v² and hence we can substitute the values .

K = 1/2 m v²

⇒ v² = 2 K/m ------(i)

Momentum = m v

⇒ p = m v

⇒ v = p/m

⇒ v² = p²/m² ---------(ii)

From (i) and (ii) we have :

p²/m² = 2 K/m

⇒ 2 K = p²/m

⇒ 2 K m = p²

⇒ p = √( 2 K m )

Let the masses be m₁ and m₂ .

p₁ : p₂ = ( √2 K₁ m₁ ) / ( √2 K₂ m₂ )

⇒ p₁ : p₂ = 1 since p₁ = p₂ .

⇒ √2 K₁ m₁ = √2 K₂ m₂

⇒ K₁ m₁ = K₂m₂

⇒ 4 K₂ m₁ = K₂ m₂ [ from 1 ]

⇒ 4 m₁ = m₂

⇒ m₁/m₂ = 1/4

⇒ m₁ : m₂ = 1 : 4

The ratio will be 1 : 4 .