Question

An object with mass $ 500 \, g $ moves along x-axis with speed $ v = 4\sqrt{x} \, m/s $. The force acting on the object is :

• A. 8 N
• B. 5 N
• C. 6 N
• D. 4 N

Hint:

When the velocity is given as a function of position, use the chain rule for differentiation to find the acceleration: \( a = v \frac{dv}{dx} \). Then, apply Newton's second law \( F = ma \) to find the force.

Answer 1

The Correct Option is D

Step 1: Convert the mass to SI units.
The mass of the object is \( m = 500 \, g = 0.5 \, kg \). 
Step 2: Find the acceleration of the object.
The speed of the object is given by \( v = 4\sqrt{x} \). 
To find the acceleration \( a \), we use the chain rule: \[ a = \frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt} = v \frac{dv}{dx} \] First, find \( \frac{dv}{dx} \): \[ \frac{dv}{dx} = \frac{d}{dx} (4\sqrt{x}) = 4 \cdot \frac{1}{2\sqrt{x}} = \frac{2}{\sqrt{x}} \] Now, substitute \( v \) and \( \frac{dv}{dx} \) into the expression for acceleration: \[ a = (4\sqrt{x}) \cdot \left(\frac{2}{\sqrt{x}}\right) = 8 \, m/s^2 \] The acceleration of the object is constant and equal to \( 8 \, m/s^2 \) along the x-axis. 
Step 3: Calculate the force acting on the object using Newton's second law.
According to Newton's second law of motion, the force \( F \) acting on an object is equal to the product of its mass \( m \) and its acceleration \( a \): \[ F = m \cdot a \] Substituting the values of mass and acceleration: \[ F = (0.5 \, kg) \cdot (8 \, m/s^2) = 4 \, kg \cdot m/s^2 = 4 \, N \] The force acting on the object is \( 4 \, N \).