Let's analyze the concept of the rate of change of momentum of an object and how it relates to forces acting on it. This fundamental principle is described by Newton's Second Law of Motion.

Understanding Momentum and its Rate of Change

Momentum is a measure of the mass and velocity of an object. It is a vector quantity, meaning it has both magnitude and direction. The momentum (\(p\)) of an object is defined as the product of its mass (\(m\)) and its velocity (\(v\)):

\(p = mv\)

The rate of change of momentum refers to how quickly the momentum of an object is changing over time. Mathematically, this can be represented as \(\frac{\Delta p}{\Delta t}\) for an average rate over a time interval \(\Delta t\), or \(\frac{dp}{dt}\) for the instantaneous rate of change.

Newton's Second Law of Motion and Rate of Change of Momentum

Newton's Second Law of Motion provides a direct relationship between the net force acting on an object and the rate at which its momentum changes. The law states that the net external force (\(F_{net}\)) applied to an object is directly proportional to the rate of change of its momentum and occurs in the same direction as the net force.

In mathematical terms, this is expressed as:

\(F_{net} \propto \frac{dp}{dt}\)

In most common forms, especially when using SI units, this proportionality becomes an equality:

\(F_{net} = \frac{dp}{dt}\)

This equation tells us that the resultant force (or net force) acting on an object is equal to the rate of change of its momentum. Therefore, the rate of change of momentum is directly proportional to the resultant force applied.

Analyzing the Options

Let's evaluate each option based on our understanding of Newton's Second Law:

• Option 1: Equal to displacement of the object
  Displacement is the change in position of an object. While forces can cause displacement by changing velocity (and thus momentum), the rate of change of momentum is not equal to the displacement itself. Displacement is a measure of position change, while the rate of change of momentum is related to the force causing the change in motion. This option is incorrect.
• Option 2: Directly proportional to the resultant force applied
  As stated by Newton's Second Law (\(F_{net} = \frac{dp}{dt}\)), the net force is equal to the rate of change of momentum. This means the rate of change of momentum is directly proportional to the resultant force. If the resultant force increases, the rate of change of momentum increases proportionally, and vice versa. This option aligns perfectly with Newton's Second Law.
• Option 3: Equal to its mass
  The rate of change of momentum depends on the force, not just the mass of the object. While mass is part of momentum (\(p=mv\)), the rate of change \(\frac{dp}{dt}\) is determined by the applied force, not simply equal to the object's mass. This option is incorrect.
• Option 4: Inversely proportional to the resultant force applied
  Inverse proportionality means that as the resultant force increases, the rate of change of momentum decreases, or vice versa. This contradicts Newton's Second Law, which states they are directly proportional. This option is incorrect.

Based on Newton's Second Law, the rate of change of momentum of an object is directly proportional to the resultant force applied to it.

Revision Table: Key Concepts

Additional Information: Relating F=ma to Rate of Change of Momentum

The more commonly known form of Newton's Second Law, \(F_{net} = ma\), where \(a\) is acceleration, is actually a special case derived from \(F_{net} = \frac{dp}{dt}\) when the mass of the object is constant.

If mass (\(m\)) is constant, then:

\(F_{net} = \frac{d(mv)}{dt}\)

Since \(m\) is constant, we can take it out of the differentiation:

\(F_{net} = m \frac{dv}{dt}\)

We know that acceleration (\(a\)) is the rate of change of velocity (\(a = \frac{dv}{dt}\)). Substituting this gives:

\(F_{net} = ma\)

So, \(F=ma\) is valid for objects with constant mass, while \(F_{net} = \frac{dp}{dt}\) is a more general form that applies even when mass changes (though this is less common in introductory physics).