Problem 36 - Physics for Scientists and Engineers with Modern Physics

Question

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

Step-by-Step Solution

Answer

Dimensions: \( A \text{ is } \text{[L/T}^4\), B \text{ is } \text{[L/T}^2\).\newline SI Units: \( A \text{ is } \text{m/s}^4, B \text{ is } \text{m/s}^2 \).

Step 1: Understand the equation

The given equation for speed is \( v = A t^3 - B t \), where \( v \) is the velocity, \( A \) and \( B \) are constants, and \( t \) is time. We need to find the dimensions and SI units of \( A \) and \( B \).

Step 2: Analyze the dimensions of velocity

Velocity \( v \) has the dimension of \( \text{[velocity]} = \text{[distance]} / \text{[time]} = \text{L} / \text{T} \).

Step 3: Determine dimensions of constant A

In the term \( A t^3 \), the dimension must be equal to that of velocity \( v \). Since \( t^3 \) represents time cubed \( \text{T}^3 \), the dimension of \( A \) must be \( \text{L} / \text{T}^4 \) so that \( [A t^3] = [A] \cdot [t^3] = (\text{L} / \text{T}^4) \cdot \text{T}^3 = \text{L} / \text{T} \).

Step 4: Determine dimensions of constant B

In the term \( B t \), the dimension must also be equal to that of velocity \( v \). Since \( t \) has dimension \( \text{T} \), \( B \) must have dimensions \( \text{L} / \text{T}^2 \) so that \( [B t] = [B] \cdot [t] = (\text{L} / \text{T}^2) \cdot \text{T} = \text{L} / \text{T} \).

Step 5: Determine SI units of A

The dimensions \( \text{L} / \text{T}^4 \) correspond to meters per seconds to the fourth power, which is \( \text{m/s}^4 \).

Step 6: Determine SI units of B

The dimensions \( \text{L} / \text{T}^2 \) correspond to meters per second squared, which is \( \text{m/s}^2 \).

Key Concepts

Velocity Equation

The velocity equation provided in the exercise is expressed as \( v = A t^3 - B t \). This equation helps us understand the relationship between time and velocity for an object.
In this context, the constants \( A \) and \( B \) are crucial as they influence how velocity changes over time. This kind of equation might be seen in physics when analyzing non-uniform motion, where velocity does not stay constant but varies with time.
By using dimensional analysis, you can determine the characteristics that these constants represent. It's essential for finding out how these constants affect the physical world and to express their meanings using standard units. This is a great example of how algebriac representations are applied to physical phenomena.

SI Units

SI Units, or the International System of Units, is a crucial part of interpreting and understanding scientific data. This system uses a universally accepted set of measurements to describe different quantities in physics.
Here's why SI Units are vital:

• They allow for consistency and accuracy when measuring and comparing different physical phenomena.
• Using SI Units ensures that experiments and results can be replicated and understood globally, facilitating communication and validation of scientific research.

The constants \( A \) and \( B \) in the equation are expressed in SI Units. For instance, the units for \( A \) were found to be \( \text{m/s}^4 \), indicating the impact of time on velocity over the fourth power.
Meanwhile, \( B \) has units of \( \text{m/s}^2 \), which is simple meters per second squared, typically related to acceleration. These units help clarify the nature of the forces or motion being described.

Physics Constants

In physics, constants like \( A \) and \( B \) in the velocity equation represent fixed quantities that determine the specific behavior of a system. Understanding these constants is integral for fully capturing the dynamics at play.
Physics constants can:

• Indicate important properties such as speed or acceleration.
• Help to predict future states of motion when initial conditions and time are known.
• Offer insights into potential energy changes and forces acting on bodies in motion.

For the equation \( v = A t^3 - B t \), \( A \) modifies how time cubed impacts velocity, potentially relating to external forces or conditions that affect motion over long durations.
The constant \( B \), associated with simple linear time, typically signifies steady or predictable changes like constant acceleration due to gravity or other forces.
By exploring these constants within dimensional analysis, students better understand the underlying principles regulating motion and its representation through equations.