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

To determine the force acting on the object, we begin by identifying the given parameters and applying relevant physics principles.

1. The object's mass is given as \(500 \, \text{g}\). For calculations in the SI unit system, this is converted to kilograms:

\(m = 0.5\, \text{kg}\)

1. The object's speed is described by the equation \(v = 4\sqrt{x}\), where \(x\) is its position on the x-axis.
2. Newton's second law of motion is used to find the force acting on the object:

\(F = m \cdot a\)

1. Acceleration \(a\), the derivative of velocity \(v\) with respect to time \(t\), must be determined. Since \(v\) is a function of \(x\), the chain rule is employed to express acceleration in terms of \(x\).
2. The derivative of \(v\) with respect to \(x\) is computed:

\(\frac{dv}{dx} = \frac{d}{dx}(4\sqrt{x}) = \frac{4}{2\sqrt{x}} = \frac{2}{\sqrt{x}}\)

1. Using the chain rule, acceleration \(a\) is expressed as the derivative of velocity with respect to time:

\(a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = \frac{dv}{dx} \cdot v = \left(\frac{2}{\sqrt{x}}\right) \cdot (4\sqrt{x}) = 8\)

1. The acceleration is found to be \(a = 8 \, \text{m/s}^2\).
2. Applying Newton's second law:

\(F = m \cdot a = 0.5 \cdot 8 = 4 \, \text{N}\)

1. Therefore, the force acting on the object is \(4 \, \text{N}\).