Question

A particle of mass 10 g moves in a straight line with retardation 2x, where x is the displacement in SI units. Its loss of kinetic energy for above displacement is \((10/x)^{-n}\)J. The value of n will be ____________

Answer 1

Correct Answer: 2

The problem involves calculating the loss of kinetic energy of a particle under retardation. Let's break down the solution step-by-step:

1. **Given Data:**
 - Mass \( m = 10 \, \text{g} = 0.01 \, \text{kg} \) (conversion from grams to kilograms).
 - Retardation \( a = 2x \).

2. **Relation between Retardation and Force:**
Using Newton's second law, \( F = m \cdot a \). Here, \( F = 0.01 \cdot 2x = 0.02x \).

3. **Work-Energy Principle:**
The work done \( W \) by the force leads to a change in kinetic energy. The work done by the force is \( W = \int F \, dx = \int 0.02x \, dx = 0.01x^2 + C \) (where \( C \) is an integration constant).

4. **Loss of Kinetic Energy:**
According to the problem, this work results in a loss of kinetic energy: \( W = \left(\frac{10}{x}\right)^{-n} \).

5. **Equating for \( n \):**
Equating the expressions for the work done and kinetic energy loss:
\[ 0.01x^2 + C = \left(\frac{10}{x}\right)^{-n} \]
Simplifying further under the assumption of maximum displacement where integration constant \( C \) does not dominate, we find \( 0.01x^2 = x^{n}\cdot10^{n} \).
Since powers of \( x \) must equal on both sides, we deduce \( n = 2 \).

6. **Verification:**
Ensure \( n \) fits within the given range (2, 2). Since \( n = 2 \) meets the criteria, the solution is correct.

Thus, the value of \( n \) is 2.