Question

Velocity can be related to acceleration and distance by the following equation: \(v^{2}=2 a x^{p}\). Find the power \(p\) that makes this equation dimensionally consistent.

Short answer

The power \(p\) is 1.

Step-by-step solution

Identify Dimensions

The dimensions of velocity \(v\) are \([L][T]^{-1}\), acceleration \(a\) are \([L][T]^{-2}\), and distance \(x\) are \([L]\). To ensure dimensional consistency, both sides of the equation \(v^{2}=2 a x^{p}\) must have the same dimensions.

Analyze dimensions of left side

The left side of the equation is \(v^{2}\). Since \(v\) has dimensions \([L][T]^{-1}\), \(v^{2}\) has dimensions \([L]^{2}[T]^{-2}\).

Analyze dimensions of right side

The right side of the equation is \(2 a x^{p}\). Ignoring the dimensionless constant 2, the dimensions are determined by \(a\) and \(x^{p}\). Since \(a\) has dimensions \([L][T]^{-2}\), \(a x^{p}\) has dimensions \([L][T]^{-2}][L]^{p}=[L]^{1+p}[T]^{-2}\).

Equate dimensions

For dimensional consistency, equate the dimensions of both sides: \([L]^{2}[T]^{-2} = [L]^{1+p}[T]^{-2}\).

Solve for power \(p\)

Since the time dimensions \([T]^{-2}\) match, equate the length dimensions: \([L]^{2} = [L]^{1+p}\). This gives the equation \(2 = 1 + p\). Solve for \(p\) to find \(p = 1\).

Key concepts

Velocity

Velocity is a fundamental concept in physics that describes how quickly an object moves in a particular direction. It is a vector quantity, which means it has both magnitude and direction. This makes velocity different from speed, which only has magnitude.

To understand velocity, consider a car moving north at 60 km/h. This means the car's velocity is 60 km/h north. The units typically used for velocity are meters per second (m/s) in the metric system, but they can vary based on context.

In dimensional analysis, the dimensions of velocity are represented as \([L][T]^{-1}\), where \([L]\) stands for length and \([T]\) for time. This showcases that velocity is the rate of change of distance (length) with respect to time.

• Velocity tells us how fast an object changes its position.
• The dimensional formula helps in equating or comparing other physical quantities involving velocity.
• It's crucial for understanding other related concepts such as acceleration and momentum.

Acceleration

Acceleration is the rate of change of velocity over time. When an object speeds up, slows down, or changes direction, it is experiencing acceleration. Like velocity, acceleration is a vector quantity, which means it includes both magnitude and direction.

To picture acceleration, imagine a car that is speeding up. If the car's velocity changes from 20 m/s to 30 m/s in 5 seconds, its acceleration is 2 m/s². This constant rate of velocity change is key in many physics problems.

In dimensional analysis, the dimensions for acceleration are written as \([L][T]^{-2}\). Here, the \([T]^{-2}\) indicates that acceleration considers time squared in the denominator, emphasizing how velocity changes over time.

• Acceleration can be positive (speeding up) or negative (slowing down), also known as deceleration.
• It helps in analyzing motion comprehensively, as seen in Newton's laws.
• The dimension gives insight into its relation with velocity and time.

Distance

Distance is a measure of how much ground an object has covered, regardless of its starting or ending point. Unlike velocity and acceleration, distance is a scalar quantity, which means it has only magnitude and no direction.

For example, if you jog around a 400-meter track twice, your distance covered is 800 meters. Even though you've returned to your starting point, the distance measures the actual path traveled.

In terms of dimensions, distance is expressed simply as \([L]\), representing length. This straightforward dimensionality makes distance a fundamental part of many physical equations.

• Distance helps quantify the total path travelled by an object.
• Unlike displacement, it does not consider direction, making it a simpler measure.
• The dimension \([L]\) is a building block for other quantities like velocity and acceleration when paired with time.