How to solve Fractional Kinetic Energy Loss in NEET — Step-by-step

Numerical/problem-solving

1. TITLE

How to solve Fractional Kinetic Energy Loss in NEET — Step-by-step

2. PROBLEM TYPE INTRO

This category of numerical deals with 1D elastic collisions where a moving body strikes a stationary target, a highly common concept in NEET. We need the formulas for fractional kinetic energy retained and transferred. The typical mistake students make is confusing the fraction of kinetic energy retained with the fraction lost (transferred to the target).

3. MCQ (Full numerical problem)

Question:
A neutron of mass $1.0 \text{ u}$ moving with an initial speed $v \text{ m/s}$ makes a head-on elastic collision with a stationary deuterium nucleus of mass $2.0 \text{ u}$. What is the fractional kinetic energy lost by the neutron?

Options:
(a) $8/9$ (Correct)
(b) $1/9$ (Common mistake: Calculating fractional energy retained instead of lost)
(c) $8/3$ (Calculation mistake: Forgetting to square the denominator)
(d) $-1/3$ (Sign error: Subtracting masses incorrectly in a derived formula)

4. STEP-BY-STEP SOLUTION

SOLUTION
Given:
* Mass of incoming particle (neutron), $m_1 = 1.0 \text{ u}$
* Mass of stationary target (deuterium), $m_2 = 2.0 \text{ u}$
* Initial speed of $m_1 = v \text{ m/s}$
* Initial speed of $m_2 = 0 \text{ m/s}$

To Find:
* Fractional kinetic energy lost by the neutron ($f_2$).

Formula:
* Fractional KE Lost: $f_2 = \frac{4m_1m_2}{(m_1 + m_2)^2}$
* Explain when to use this formula: Use this direct derived formula when you need to find the proportion of initial kinetic energy that is transferred from a moving mass to a stationary mass in a completely elastic 1D collision.

Step 1: Identify and assign the mass values.
$m_1 = 1.0 \text{ u}$ and $m_2 = 2.0 \text{ u}$.

Step 2: Substitute the values into the fractional energy lost formula.
$f_2 = \frac{4(1.0 \text{ u})(2.0 \text{ u})}{(1.0 \text{ u} + 2.0 \text{ u})^2}$

Step 3: Simplify the numerator and the denominator.
$f_2 = \frac{8.0 \text{ u}^2}{(3.0 \text{ u})^2}$
$f_2 = \frac{8.0}{9.0}$

Answer: $8/9$ (Unitless fraction)

5. WHERE STUDENTS GO WRONG

List 2-3 common mistakes:
- ❌ Mistake 1: Calculating retained energy instead of lost energy. Students often calculate $f_1 = (\frac{m_1 - m_2}{m_1 + m_2})^2$ (which gives $1/9$), forgetting that the question asked for the energy lost to the target, which is $1 - f_1$.
- ❌ Mistake 2: Forgetting to square the denominator. In the rush of the exam, students will calculate $\frac{4m_1m_2}{m_1 + m_2}$, completely missing the crucial "squared" on the total mass, leading to dimensionally incorrect answers.
- ❌ Mistake 3: Wasting time calculating individual final velocities. Calculating final velocity $v_{1f}$ using long equations and then doing $\frac{1}{2}m_1v_{1f}^2$ manually burns precious minutes. Always use the direct ratio formula!

6. QUICK FORMULA REFERENCE

Formulas Used:

• Formula 1 — Use when finding Fractional KE Lost (Transferred): $f_2 = \frac{4m_1m_2}{(m_1+m_2)^2}$. Use this when calculating the energy the incoming mass transfers to a stationary target.
• Formula 2 — Use when finding Fractional KE Retained: $f_1 = (\frac{m_1 - m_2}{m_1 + m_2})^2$. Use this when calculating how much energy the original moving mass keeps after bouncing off the target.

7. SIMILAR PROBLEM HINT

"If NEET changes the values, the method stays the same. Practice with: A neutron colliding elastically with a stationary Carbon nucleus ($m_2 = 12.0 \text{ u}$) to find the fraction of energy transferred."

8. NCERT REFERENCE

Based on: NCERT Class 11, Chapter 5, Example 5.11 (Slowing down of neutrons)

9. CTA

More solved numericals → Work, Energy and Power Complete Guide
Practice problems → [Topic Test: Work, Energy and Power]