A body of mass 500 g moves along $$x$$-axis such that it's velocity varies with displacement $$x$$ according to the relation $$v = 10\sqrt{x}$$ m s$$^{-1}$$. The force acting on the body is:

A body of mass 500 g moves along the x-axis with velocity $$v = 10\sqrt{x}$$ m/s. We need to find the force acting on it.

First, we state the given quantities.

Mass: $$m = 500$$ g $$= 0.5$$ kg

Velocity as a function of displacement: $$v = 10\sqrt{x} = 10x^{1/2}$$ m/s

Next, we find the acceleration using the chain rule.

Since velocity is given as a function of position (not time), we use the relation:

$$ a = v\frac{dv}{dx} $$

This follows from the chain rule: $$a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = \frac{dv}{dx} \cdot v$$.

First, find $$\frac{dv}{dx}$$:

$$ \frac{dv}{dx} = \frac{d}{dx}(10x^{1/2}) = 10 \times \frac{1}{2}x^{-1/2} = \frac{5}{\sqrt{x}} $$

Now compute the acceleration:

$$ a = v \cdot \frac{dv}{dx} = 10\sqrt{x} \times \frac{5}{\sqrt{x}} = 50\;\text{m/s}^2 $$

Notice that the $$\sqrt{x}$$ terms cancel, giving a constant acceleration independent of position.

From this, we apply Newton's second law.

$$ F = ma = 0.5 \times 50 = 25\;\text{N} $$

The force acting on the body is 25 N.

The correct answer is Option 2: 25 N.