Problem 36 - Physics for Scientists and Engineers with Modern Physics

Question

The speed \(v\) of an object is given by the equation \(v=A t^{3}-B t,\) where \(t\) refers to time. \((a)\) What are the dimensions of \(A\) and \(B ?(b)\) What are the SI units for the constants \(A\) and \(B ?\)

Step-by-Step Solution

Answer

(a) Dimensions: \( [A] = [LT^{-4}] \), \( [B] = [LT^{-2}] \). (b) Units: \( A \) in \( \text{m/s}^4 \), \( B \) in \( \text{m/s}^2 \).

Step 1: Analyzing the Given Equation

We have the equation for speed: \( v = A t^3 - B t \). Since speed has the dimension of \([LT^{-1}]\), we will match the dimensions of all terms on the right-hand side to \([LT^{-1}]\).

Step 2: Finding the Dimension of A

Consider the term \( A t^3 \). The dimension of this term should equal the dimension of speed \([LT^{-1}]\). So, \([A][T]^3 = [LT^{-1}]\). Solving for \([A]\), we get \([A] = [LT^{-1}][T]^{-3} = [LT^{-4}]\).

Step 3: Finding the Dimension of B

Now consider the term \( B t \). Similarly, \([B][T] = [LT^{-1}]\), which gives \([B] = [LT^{-1}][T]^{-1} = [LT^{-2}]\).

Step 4: Determining SI Units for A

From the dimension \([A] = [LT^{-4}]\), the SI units for \(A\) are obtained by substituting \(L\) as meters (m) and \(T\) as seconds (s). So, \(A\) has the units \( \text{m/s}^4 \).

Step 5: Determining SI Units for B

Using the dimension \([B] = [LT^{-2}]\), we substitute \(L\) as meters and \(T\) as seconds. So, \(B\) has units \( \text{m/s}^2 \).

Key Concepts

SI Units

SI Units, or the International System of Units, provide a standardized method of measurement that is used around the globe. This system aids in maintaining consistency and precision in scientific calculations and everyday measurements. For example:

• Length is measured in meters (m).
• Time is measured in seconds (s).
• Mass is measured in kilograms (kg).

For speed, which is determined using the equation provided in the exercise, the SI units are meters per second (m/s). By establishing these units, it ensures that measurements can be universally understood and compared.

Speed Equation

In physics, speed is an essential concept defined as the rate at which an object covers distance. The speed equation in the exercise is given as \( v = A t^3 - B t \). Here:

• \(v\) represents speed.
• \(A t^3\) and \(B t\) are terms that help determine speed at any given time \(t\).

Understanding the specific role of each term can illuminate how different factors affect speed. In more complex scenarios, the difference between these terms \(A t^3\) and \(B t\) could signify acceleration impacts or other forces at play.

Unit Conversion

Unit conversion is vital for transforming one measurement system into another to ease understanding and calculations. In the context of the given equation, unit conversion helps in interpreting the constants \(A\) and \(B\) in recognizable units.To find the SI units for \(A\) and \(B\), we use dimensional analysis:

• For \(A\), with the dimension \([LT^{-4}]\), the SI unit is meters per second to the fourth \((\text{m/s}^4)\).
• For \(B\), which has the dimension \([LT^{-2}]\), its SI unit is meters per second squared \((\text{m/s}^2)\).

Whenever you encounter formulas like these, understanding the dimensions and converting them accurately helps not just in grasping theoretical concepts but also in applying them in real-world scenarios where units play a crucial role.