Question

Acceleration is related to distance and time by the following equation: \(a=2 x t^{p}\). Find the power \(p\) that makes this equation dimensionally consistent.

Short answer

Power \(p\) is \(-2\).

Step-by-step solution

Define Dimensional Formula of Acceleration

The dimensional formula for acceleration is given by \([a] = [L][T]^{-2}\). Here, \([L]\) represents length (or distance), and \([T]\) represents time.

Define Dimensional Formula of Distance and Time

The dimensional formula for distance \(x\) is \([L]\), as it is a measure of length. The dimensional formula for time \(t\) is \([T]\).

Substitute Dimensions into the Given Equation

Substitute the dimensions into the equation \(a = 2xt^{p}\). Therefore, the dimensional equation is \[[L][T]^{-2} = [L][T]^{p}\]

Solve for Power \(p\)

To find \(p\), compare the powers of \([L]\) and \([T]\) on both sides of the equation. For \([L]\), the powers already match, as both are \(1\). For \([T]\), we have \(-2 = p\). Therefore, \(p = -2\).

Key concepts

Acceleration

Acceleration is a key concept in physics that relates to how quickly an object's velocity changes with time. It's a measure that tells us how much faster or slower something is moving based on the change in its velocity over a period of time. This is expressed in units of distance over time squared, such as meters per second squared (\( m/s^2 \)).
By understanding acceleration, we can predict future positions of objects, calculate forces, and design systems that operate safely within expected ranges. It is often used when calculating the dynamics of vehicles, for instance, to determine how quickly they can come to a stop or reach a certain speed.
In mathematical terms, acceleration (\(a\)) is often represented as:

• Forward acceleration: Positive values indicate increasing speed.
• Negative acceleration (Deceleration): Negative values suggest a reduction in speed.

Thus, whenever studying problems involving movement, recognizing acceleration's dimensionality and formula can provide valuable insights.

Dimensional Formula

A dimensional formula is a tool that helps express the physical quantity in terms of basic dimensions like Length (\([L]\)), Mass (\([M]\)), and Time (\([T]\)). This approach standardizes the representation of different physical quantities, aiding in identifying relationships and ensuring consistency in equations.
Taking the example from our exercise, the dimensional formula for acceleration is \([L][T]^{-2}\). This indicates that acceleration is indirectly related to time and directly related to distance.
The main benefits of dimensional formulas include:

• They help in converting units from one system to another.
• They assist in checking the correctness of physical equations (dimensional analysis).
• They provide a method to derive relations between different physical quantities.

Dimensional formulas are fundamental in ensuring that equations make sense "dimension-wise", playing an integral role in understanding and verifying scientific and engineering problems.

Distance and Time Relationship

Distance and time are two fundamental aspects when analyzing motion. In physics, they describe how objects move through space over a given period. Understanding their relationship is crucial for solving problems involving velocity, acceleration, or displacement.
In our particular equation, the relationship is expressed as \(x t^p\), highlighting how distance (\(x\)) and a power of time (\(t^p\)) interplay to yield certain properties of motion, such as acceleration. Here, the task was to find the power \(p\) that maintains dimensional consistency. As deduced, \(p = -2\), suggesting that time inversely affects the acceleration.
Some important insights into the distance and time relationship are:

• They provide a framework to understand how fast or slowly events unfold over time.
• The power relationship can suggest dynamics such as oscillations and decay.
• Direct relationships imply proportional changes, whereas inverse relationships indicate dilation or compression over time.

Mastering the interaction between distance and time sets the groundwork for exploring complex physical phenomena and solving practical physics problems effectively.