Question

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

Short answer

The power \(p\) is 1.

Step-by-step solution

Identify Dimensions of Each Variable

Identify the dimensions of acceleration \(a\), velocity \(v\), and time \(t\). Acceleration \(a\) has dimensions \([L][T]^{-2}\), velocity \(v\) has dimensions \([L][T]^{-1}\), and time \(t\) has dimensions \([T]\).

Write the Dimensional Equation

The equation is \(a = vt^{p}\). We need to express each term in terms of dimensions: \([L][T]^{-2} = [L][T]^{-1}[T]^{p}\).

Solve for the Power \(p\)

Match the dimensions on both sides of the equation. The left side is \([L][T]^{-2}\) and the right side is \([L][T]^{-1+p}\). For the dimensions to be consistent, we equate the powers of \(L\) and \(T\):\([-1 + p = -2\).

Equate Powers of Dimensions and Solve for \(p\)

From \([-1 + p = -2\), solving for \(p\) gives \(p = 1\). This ensures dimensional consistency between both sides of the equation.

Key concepts

Physics Problem Solving

When it comes to physics problem solving, one of the most fundamental tools in your toolkit is dimensional analysis. This involves checking the consistency of units across the terms in an equation. With dimensions, you can determine if an equation makes sense before getting into the nitty-gritty of calculations. This is particularly useful in exercises where you're asked to find unknown variables or powers, like the power of time in a given acceleration equation. Dimensional analysis allows you to systematically approach the problem by comparing the dimensions of all physical quantities involved. This ensures that each side of the equation matches in terms of their fundamental units, like length, time, and mass.

• Identify the units of each variable
• Create a dimensional equation
• Equate the dimensions to find unknowns or check consistency

Break down the problem into smaller, more manageable parts, and focus on how each part contributes to the whole solution.

Acceleration

Acceleration is an essential concept in physics and relates to how quickly an object changes its velocity. It is described as the rate of change of velocity per unit of time. The dimensions of acceleration are \( [L][T]^{-2} \), where \( L \) is for length and \( T \) is for time. Essentially, this tells us how much the velocity (which itself is a change in position) changes every second.

• Acceleration is a vector, meaning it has both magnitude and direction.
• It can be caused by changes in the speed or direction of an object.
• If an object speeds up, slows down, or changes direction, it is accelerating.

Understanding acceleration is crucial for solving many types of physics problems, especially those involving motion and forces.

Velocity

Velocity is another fundamental physics concept that describes the rate at which an object changes its position. Not to be confused with speed, velocity is a vector quantity, meaning it includes both magnitude and direction. If you're traveling north at 60 km/h, your velocity is distinct from someone going south at the same speed due to the change in direction. The dimensions of velocity are given by \( [L][T]^{-1} \), which encapsulates how much distance is covered over a unit of time.

• Velocity tells you how fast something is moving and in which direction.
• It can be constant or change over time, depending on whether there's an acceleration present.
• Average velocity is different from instantaneous velocity; the former is over a time interval, the latter at a specific moment.

Establishing a clear understanding of velocity is important, as it interlinks with acceleration and time in motion equations.

Time

Time is a universal component in measuring motion, expressed in dimensions as \([T]\). It's the independent variable in many physics equations, representing the duration over which events occur or changes take place. In the context of motion, time helps describe how quickly or slowly these events happen.

• Time is a scalar quantity, meaning it only has magnitude, not direction.
• It plays a crucial role in defining rates like velocity and acceleration.
• Understanding time's role in equations helps in predicting and analyzing the behavior of moving objects.

Often, when solving physics problems, you'll manipulate time alongside other variables to understand how one influences the other over a given duration.