Question

Which of the following expressions correctly illustrates the SI base units for each of the variables in the formula below? $$m \Delta \mathbf{v}=\mathbf{F} \Delta t$$ A \(\mathrm{lb} \times \mathrm{mph}=\mathrm{ft} \times \mathrm{lb} \times \mathrm{s}\) B \(\mathrm{kg} \times \frac{\mathrm{m}}{\mathrm{s}}=\mathrm{N} \times \mathrm{s}\) C \(\mathrm{kg} \times \frac{\mathrm{m}}{\mathrm{s}}=\frac{\mathrm{kg} \cdot \mathrm{m}}{\mathrm{s}^{2}} \times \mathrm{s}\) D \(g \times \frac{m}{s}=\frac{g \cdot m}{s^{2}} \times s\)

Short answer

B: \(\mathrm{kg} \times \frac{\mathrm{m}}{\mathrm{s}}=\mathrm{N} \times \mathrm{s}\).

Step-by-step solution

Identify SI units for mass and velocity change

The variable 'm' represents mass and in the SI unit system, mass is measured in kilograms (kg). The change in velocity, \( \Delta \mathbf{v} \), is velocity measured in meters per second (m/s). Therefore, the left-hand side of the equation \( m \Delta \mathbf{v} \) becomes \( \mathrm{kg} \times \frac{\mathrm{m}}{\mathrm{s}} \).

Identify SI units for force and time change

The variable \( \mathbf{F} \) represents force. In the SI unit system, force is measured in newtons (N), where one newton is equivalent to \( \frac{\mathrm{kg} \cdot \mathrm{m}}{\mathrm{s}^2} \). The variable \( \Delta t \) represents the change in time and is measured in seconds (s).

Combine the units on the right-hand side of the equation

Combine the SI units for force \( \mathbf{F} \) and time \( \Delta t \). The right-hand side of the equation \( \mathbf{F} \Delta t \) becomes \( \frac{\mathrm{kg} \cdot \mathrm{m}}{\mathrm{s}^2} \times \mathrm{s} \). When simplified, this results in units of \( \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} \).

Match expressions with given options

Now, we compare the simplified units from both sides of the equation with the given options: \ (\mathrm{kg} \times \frac{\mathrm{m}}{\mathrm{s}}=\mathrm{N} \times \mathrm{s}) \ and \ (\mathrm{kg} \times \frac{\mathrm{m}}{\mathrm{s}}=\frac{\mathrm{kg} \cdot \mathrm{m}}{\mathrm{s}^{2}} \times \mathrm{s}) \ both match. But we must choose the form that directly shows the simplified version: \( \mathrm{kg} \times \frac{\mathrm{m}}{\mathrm{s}}=\mathrm{N} \times \mathrm{s} \).

Key concepts

Newton's Second Law

Newton's Second Law of Motion is a fundamental principle in physics that defines the relationship between the motion of an object and the forces acting on it. The law is commonly written as \(\textbf{F} = m \textbf{a}\), where \(\textbf{F}\) is the force applied, \(\textbf{m}\) is the mass of the object, and \(\textbf{a}\) is the acceleration of the object. In this context, force causes an object to accelerate, proportionally to the object's mass. In simpler terms, for a given force, a heavier object will accelerate less than a lighter one.
In the exercise \(\textbf{m} \Delta \textbf{v} = \textbf{F} \Delta t\), Newton's Second Law can be seen implicitly within the formula's setup, showing the change in momentum (\

SI Units

The International System of Units (SI) is the modern form of the metric system. It is the world's most widely used system for measurement, providing a consistent framework for scientific research and applications. SI Units are standardized, ensuring that measurements are universally understood and comparable.
Here are some key SI units relevant to the exercise:
- Mass \(\textbf{m}\) is measured in kilograms (kg).
- Distance is measured in meters (m).
- Time is measured in seconds (s).
- Force \(\textbf{F}\) is measured in newtons (N), which is equivalent to \(\frac{kg \cdot m}{s^{2}} \).
These units are crucial for performing dimensional analysis and ensuring equations are dimensionally consistent.

Dimensional Analysis

Dimensional analysis is a technique used in physics and engineering to check the consistency of equations by comparing the dimensions on both sides of the equation. It ensures that equations make sense in terms of physical units.
In the given problem, dimensional analysis helps verify whether the units on both sides of the equation \(\textbf{m} \Delta \textbf{v} = \textbf{F} \Delta t\) are consistent:

• Mass \(\textbf{m}\) has units of kilograms (kg).
  
• Change in velocity \(\textbf{\Delta v}\) has units of meters per second (m/s). So, the left side of the equation becomes \(kg \cdot \frac{m}{s} \)
  
• Force \(\textbf{F}\) has units of newtons (N), which can be expressed as \(\frac{kg \cdot m}{s^{2}} \).
  
• Time \(\textbf{\Delta t}\) has units of seconds (s). So, the right side of the equation becomes \(\frac{kg \cdot m}{s^{2}} \cdot s = kg \cdot \frac{m}{s} \)
  

By comparing both sides, we find they have consistent units, confirming the equation's correctness.

Momentum

Momentum is a key concept in physics that describes the quantity of motion an object has. It is defined as the product of an object's mass and velocity, represented as \(\textbf{p} = m \textbf{v}\). Momentum is a vector quantity, meaning it has both magnitude and direction.
The exercise uses momentum in the context of the equation \(\textbf{m} \Delta \textbf{v} = \textbf{F} \Delta t\). Here, \(\textbf{m} \Delta \textbf{v}\) refers to the change in momentum, often referred to as impulse. Impulse is the change in momentum resulting from a force applied over a period of time.
Knowing the relationship between force, time, and momentum helps in understanding various physical phenomena, from car crashes to the motion of celestial bodies. By solving for momentum, one can predict how objects will move and interact under different forces.