Question

Compare the momentum of two objects of mass 10 kg and 40 kg respectively but having same kinetic energy.​

Answer 1

Answer :

• The ratio of the momentum of the two objects is 1 : 2.

Explanation :

Given :

• Mass of the first body, m₁ = 10 kg
• Mass of the second body, m₂ = 40 kg
• Kinetic energy possessed by both the bodies are equal.

To find :

• Ratio of momentum of the two bodies.

Knowledge required :

• Formula for Kinetic energy (K.E.) possessed by a body :

⠀⠀⠀⠀⠀⠀⠀⠀ ⠀K.E. = ½mv²⠀

[Where : K.E. = Kinetic energy possessed by the body; m = Mass of the body; v = Velocity with which the body is moving]

• Formula for momentum of a body

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀p = mv⠀

[Where : p = Momentum of the body; m = Mass of the body; v = Velocity of the body]

Solution :

Let the two bodies be A and B.

For the body A, let the mass of the body be m₁ and velocity of the body be v₁.

For the body B, let the mass of the body be m₂ and velocity of the body be v₂.

First let us find the value of (v₁/v₂) velocity of both the bodies.

According to the question, "The Kinetic energy possessed by the both the bodies (i.e, Body A and Body B) is equal.

Mathematically,

⠀⠀⠀⠀⠀⠀⠀⠀⠀K.E.₁ = K.E.₂⠀

[Here, K.E.₁ is for the body A and K.E.₂ is for the body B]

By using the equation for Kinetic energy possessed by a body and substituting it ghe above equation, we get :

⠀⠀=> K.E.₁ = K.E.₂

⠀⠀=> ½m₁v₁² = ½m₂v₂²

Now by substituting the values in it, we get :

⠀⠀=> ½ × 10 × v₁² = ½ × 40 × v₂²

⠀⠀=> 5 × v₁² = 20 × v₂²

⠀⠀=> 5 × v₁² = 20 × v₂²

⠀⠀=> v₁²/v₂² = 20/5

⠀⠀=> (v₁/v₂)² = 4/1

⠀⠀=> v₁/v₂ = √4/1

⠀⠀=> v₁/v₂ = 2/1

⠀⠀⠀⠀⠀⠀∴ v₁/v₂ = 2/1

Hence the value of (v₁/v₂) of the two bodies is 2/1.

Now coming to the question, we have to find,

"The ratio of the momentum of the two bodies"

We can write it mathematically as,

⠀⠀⠀⠀⠀⠀⠀⠀⠀p₁ : p₂⠀

[Here, p₁ is for the body A and p₂ is for the body B]

By using the equation for me momentum of a body and substituting it ghe above equation, we get :

⠀⠀=> p₁ : p₂

⠀⠀=> p₁ : p₂ = m₁v₁ : m₂v₂

⠀⠀=> p₁/p₂ = m₁v₁/m₂v₂

⠀⠀=> p₁/p₂ = m₁/m₂ × v₁/v₂

Now by substituting the values in it, we get :

⠀⠀=> p₁/p₂ = 10/40 × 2/1 ⠀⠀ [∴ v₁/v₂ = 2/1]

⠀⠀=> p₁/p₂ = 1/4 × 2/1

⠀⠀=> p₁/p₂ = 1/2

⠀⠀=> p₁ : p₂ = 1 : 2

Hence the ratio of the momentum of the two bodies is 1 : 2.