Question

If the kinetic energy of a particle is increased by 44%, the increase in its linear momentum will be
1. 20%
2. 40%
3. 10%
4. 15%

Answer 1

Correct Answer - Option 1 : 20%

The correct answer is option 1) i.e. 20%

CONCEPT:

• ​Kinetic energy is the energy possessed by a moving object. Kinetic energy (KE) is expressed as:

\(⇒ KE =\frac{1}{2} mv^2\)

Where m is the mass of the object and v is the velocity of the object.

• Momentum: Momentum is the impact due to a moving object of mass m and velocity v.
• The momentum (p) of an object is expressed as:

⇒ p = mv

CALCULATION:

Let the initial and final kinetic energy of the particle be KE₁ and KE₂. Similarly, the linear momentum will be p₁ and p₂.

When the KE of the particle is increased by 44%, KE₂ = 1.44KE₁

We know,

\(⇒ KE =\frac{1}{2} mv^2\)

Multiplying and dividing by m we get,  

\(⇒ KE =\frac{1}{2} mv^2 \times \frac{m}{m} = \frac{p^2}{2m}\)

\(⇒ p =\sqrt{2mKE}\)

⇒ p ∝ √KE

• The percentage increase in linear momentum 

\(\Rightarrow\Delta p\%=\frac{p_2 - p_1}{p_1} \times 100 = \frac{\sqrt{KE_2} - \sqrt{KE_1}}{\sqrt{KE_1}} \times 100 \)

\(\Rightarrow \Delta p\%=\frac{\sqrt{1.44KE_1} - \sqrt{KE_1}}{\sqrt{KE_1}} \times 100 =\frac{\sqrt{1.44} - \sqrt{1}}{\sqrt{1}} \times 100 = \frac{1.2 -1}{1} \times 100 = 20\%\)