Understanding Rate of Change of Momentum and Force

The question asks about the relationship between the rate of change of momentum of an object and other quantities. This is a fundamental concept in physics, directly related to Newton's laws of motion.

What is Momentum?

Momentum ($\vec{p}$) is a measure of the mass in motion. For an object of mass $m$ moving with velocity $\vec{v}$, its momentum is given by the equation:

$$\vec{p} = m\vec{v}$$

Momentum is a vector quantity, meaning it has both magnitude and direction. The unit of momentum is typically kilogram-meter per second (kg⋅m/s).

Rate of Change of Momentum

The rate of change of momentum refers to how quickly the momentum of an object is changing over time. Mathematically, this is represented as $\frac{d\vec{p}}{dt}$.

Let's consider an object with constant mass $m$. The rate of change of momentum is:

$$\frac{d\vec{p}}{dt} = \frac{d(m\vec{v})}{dt}$$

Since $m$ is constant, we can take it out of the differentiation:

$$\frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt}$$

We know that the rate of change of velocity ($\frac{d\vec{v}}{dt}$) is the acceleration ($\vec{a}$). So, the rate of change of momentum is:

$$\frac{d\vec{p}}{dt} = m\vec{a}$$

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

Newton's Second Law of Motion states that the resultant force ($\vec{F}_{net}$) acting on an object is equal to the rate of change of its momentum. Mathematically, this is expressed as:

$$\vec{F}_{net} = \frac{d\vec{p}}{dt}$$

This equation is the most general form of Newton's Second Law. For a constant mass object, as shown above, this reduces to $\vec{F}_{net} = m\vec{a}$.

The equation $\vec{F}_{net} = \frac{d\vec{p}}{dt}$ directly tells us the relationship between the resultant force applied to an object and its rate of change of momentum.

Analyzing the Options

Let's examine the given options based on our understanding of Newton's Second Law:

1. Directly proportional to the resultant force applied
2. Equal to displacement of the object
3. Inversely proportional to the resultant force applied
4. Equal to its mass

From Newton's Second Law, $\vec{F}_{net} = \frac{d\vec{p}}{dt}$. This equation shows that the resultant force ($\vec{F}_{net}$) and the rate of change of momentum ($\frac{d\vec{p}}{dt}$) are equal vectors. If two quantities are equal, they are also directly proportional (with a constant of proportionality equal to 1).

Let's evaluate the other options:

• Option 2: Equal to displacement of the object. Displacement is a change in position, measured in meters. The rate of change of momentum is related to force, measured in Newtons. These quantities are fundamentally different and not equal.
• Option 3: Inversely proportional to the resultant force applied. This contradicts Newton's Second Law, which states they are directly proportional (or equal).
• Option 4: Equal to its mass. Mass is a scalar quantity measured in kilograms. The rate of change of momentum is a vector quantity measured in Newtons (or kg⋅m/s²). They are not equal.

Therefore, the rate of change of momentum of an object is directly proportional to the resultant force applied to it, as stated by Newton's Second Law.

Summary of the Relationship

The core relationship is given by Newton's Second Law:

This confirms that the rate of change of momentum is directly proportional to the resultant force applied.

Revision Table: Rate of Change of Momentum

Additional Information: Impulse and Momentum

The concept of rate of change of momentum is closely related to impulse. Impulse ($ \vec{J} $) is defined as the change in momentum of an object:

$$ \vec{J} = \Delta \vec{p} = \vec{p}_f - \vec{p}_i $$

where $\vec{p}_f$ is the final momentum and $\vec{p}_i$ is the initial momentum.

From Newton's Second Law, $\vec{F}_{net} = \frac{d\vec{p}}{dt}$. If a constant net force $\vec{F}_{net}$ acts over a time interval $\Delta t$, we can approximate $\frac{d\vec{p}}{dt}$ as $\frac{\Delta \vec{p}}{\Delta t}$.

So, $\vec{F}_{net} \approx \frac{\Delta \vec{p}}{\Delta t}$, which gives $\vec{F}_{net} \Delta t \approx \Delta \vec{p}$.

The term $\vec{F}_{net} \Delta t$ is defined as impulse. Thus, the impulse-momentum theorem states that the impulse applied to an object is equal to the change in its momentum:

$$ \vec{J} = \vec{F}_{net} \Delta t = \Delta \vec{p} $$

This further highlights the direct link between force (specifically, force applied over time) and the change in momentum (which is related to the rate of change of momentum).