Question

A body of mass 500 g moves along x-axis such that it's velocity varies with displacements x according to the relation \(v  =10 \sqrt x\) m/s the force acting on the body is :

• A. 25 N
• B. 125 N
• C. 5 N
• D. 166 N

Answer 1

The Correct Option is A

Given: 

• Mass (\( m \)) = \( 500 \, \text{g} = 0.5 \, \text{kg} \)
• Velocity (\( v \)) = \( 10 \sqrt{x} \, \text{m/s} \)

Step 1: Find the Acceleration

Acceleration (\( a \)) is the rate of change of velocity with respect to time, which can be expressed as:

\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = v \cdot \frac{dv}{dx}. \]

Given \( v = 10 \sqrt{x} \), differentiate \( v \) with respect to \( x \):

\[ \frac{dv}{dx} = \frac{d}{dx} (10 \sqrt{x}) = 10 \cdot \frac{1}{2 \sqrt{x}} = \frac{5}{\sqrt{x}}. \]

Now substitute \( v = 10 \sqrt{x} \) and \( \frac{dv}{dx} = \frac{5}{\sqrt{x}} \) into the formula for acceleration:

\[ a = v \cdot \frac{dv}{dx} = 10 \sqrt{x} \cdot \frac{5}{\sqrt{x}} = 50 \, \text{m/s}^2. \]

Step 2: Calculate the Force

Using Newton’s second law, the force (\( F \)) is given by:

\[ F = m \cdot a. \]

Substitute \( m = 0.5 \, \text{kg} \) and \( a = 50 \, \text{m/s}^2 \):

\[ F = 0.5 \cdot 50 = 25 \, \text{N}. \]

Final Answer:

The force acting on the body is \( 25 \, \text{N} \).