Understanding Newton's Second Law and Linear Momentum

This question asks us to complete the statement of Newton's second law of motion, specifically focusing on the relationship between force and linear momentum. Let's break down the concepts involved.

Defining Linear Momentum

First, what is linear momentum? In physics, linear momentum is a measure of an object's motion. It depends on both the object's mass and its velocity. Mathematically, momentum ($ \vec{p} $) is defined as the product of an object's mass ($m$) and its velocity ($ \vec{v} $): $ \vec{p} = m\vec{v} $ Momentum is a vector quantity, meaning it has both magnitude and direction. The direction of the momentum is the same as the direction of the velocity.

Newton's Second Law in Terms of Momentum

Newton's second law of motion provides a fundamental relationship between force and motion. While often stated as $ \vec{F} = m\vec{a} $ (Force equals mass times acceleration), a more general and precise form of the law relates force to the rate of change of linear momentum. This general form is particularly useful when dealing with situations where mass might change or when we need a deeper understanding of dynamics.

The law states that the instantaneous rate of change of the linear momentum of a particle is equal to the vector sum of all the external forces acting on it, and this change occurs in the direction of the net force.

Using calculus, this can be expressed as:

• $ \vec{F}_{\text{net}} $ represents the net external force acting on the particle.
• $ \frac{d\vec{p}}{dt} $ represents the instantaneous rate of change of linear momentum with respect to time ($t$).

Completing the Statement

Based on the formula $ \vec{F}_{\text{net}} = \frac{d\vec{p}}{dt} $, we can see that the net external force ($ \vec{F}_{\text{net}} $) is precisely equal to the rate of change of linear momentum ($ \frac{d\vec{p}}{dt} $). This means the net external force is directly proportional to (in fact, equal to) and in the same direction as the rate of change of the particle's linear momentum.

Therefore, the blank in the question should be filled with "net external force".

Analysis of Options

Let's look at why the other options are not the best fit:

• Net external force: This aligns perfectly with the definition derived from Newton's second law. It accounts for all forces acting on the object.
• Applied force: While an applied force is often a significant force acting on an object, it might not be the *only* force. Forces like friction, gravity, or air resistance also act. Newton's second law specifically deals with the sum of *all* external forces, which is the net force.
• Impulse: Impulse ($ \vec{J} $) is defined as the change in momentum ($ \Delta\vec{p} $) and is also equal to the average net force multiplied by the time interval over which it acts ($ \vec{J} = \vec{F}_{\text{net, avg}} \Delta t = \Delta\vec{p} $). While related to momentum change, impulse is not what the *rate* of change of momentum is directly proportional to; rather, the force itself is.
• Power: Power ($P$) is the rate at which work is done or energy is transferred ($ P = \frac{dW}{dt} $). It is related to force and velocity ($ P = \vec{F} \cdot \vec{v} $) but is fundamentally different from the concept of the rate of change of momentum.

Conclusion

Newton's second law of motion fundamentally states that the net external force acting on an object causes a change in its linear momentum, and the rate at which this momentum changes is directly proportional to the net external force itself.