Question

Which of the following quantities has the same dimension as a distance? A. \(v t\) B. \(\frac{1}{2} a t^{2}\) C. \(2 a t\) D. \(v^{2} / a\) Note that \(v\) is speed, \(t\) is time, and \(a\) is acceleration. Refer to Table \(1.5\) for the corresponding dimensions.

Short answer

Options A, B, and D have the same dimension as distance.

Step-by-step solution

Understand the Dimensions

To determine which quantity has the same dimension as distance, we need to understand the dimensions of the given variables: speed \(v\) has dimensions \([L][T]^{-1}\), time \(t\) has dimensions \([T]\), and acceleration \(a\) has dimensions \([L][T]^{-2}\). Distance has dimensions of \([L]\).

Analyze Option A: \(v t\)

Calculate the dimensions of \(v t\). The dimension of \(v\) is \([L][T]^{-1}\) and \(t\) is \([T]\). Thus, the dimensions of \(v t\) are \([L][T]^{-1} \times [T] = [L]\). This matches the dimension of distance.

Analyze Option B: \(\frac{1}{2} a t^2\)

Calculate the dimensions of \(\frac{1}{2} a t^2\). The dimensions of \(a\) are \([L][T]^{-2}\) and \(t^2\) is \([T]^2\). Thus, the dimensions of \(a t^2\) are \([L][T]^{-2} \times [T]^2 = [L]\). This matches the dimension of distance.

Analyze Option C: \(2 a t\)

Calculate the dimensions of \(2 a t\). The dimensions of \(a\) are \([L][T]^{-2}\) and \(t\) is \([T]\). Thus, the dimensions of \(a t\) are \([L][T]^{-2} \times [T] = [L][T]^{-1}\). This does not match the dimension of distance.

Analyze Option D: \(v^2 / a\)

Calculate the dimensions of \(v^2 / a\). The dimensions of \(v^2\) are \(([L][T]^{-1})^2 = [L]^2[T]^{-2}\). Dividing by \(a\), which is \([L][T]^{-2}\), the resulting dimensions are \([L]^2[T]^{-2} / [L][T]^{-2} = [L]\). This matches the dimension of distance.

Conclusion

From our analysis, the quantities \(v t\), \(\frac{1}{2} a t^2\), and \(v^2 / a\) have the same dimension as distance \([L]\).

Key concepts

Distance

Distance is a fundamental measure in physics that describes how far an object moves from one point to another. It is a scalar quantity, meaning it only has magnitude and no direction. In terms of dimensions, distance is represented as \[L\]. This means that it is related to length. The calculation of distance can vary depending on the context:

• For linear motion, it's the direct path between two points.
• In circular motion, it might refer to how much path is covered along the arc.

Understanding distance helps us analyze an object’s position and movement in space. It lays the foundation for our investigation into other related topics, such as speed, acceleration, and time—which are pivotal concepts when studying motion.

Speed

Speed is the rate at which an object covers distance. Unlike velocity, which is a vector and includes direction, speed is a scalar value focusing purely on how fast something is moving. The formula for speed is usually expressed as \text{speed=distance/time}\.To determine the dimensions of speed, consider that:

• Distance has the dimension \[L\], as we learned earlier.
• Time has the dimension \[T\].

The dimension of speed thus becomes \[\frac{[L]}{[T]} = [L][T]^{-1}\]. This dimensional analysis helps us reiterate that speed is concerned with how quickly an object travels over a period.

Acceleration

Acceleration occurs when there is a change in the speed or direction of an object's movement. It is a vector quantity because it involves both magnitude and direction. The formula to calculate acceleration is \text{acceleration = change in velocity / time}\.To understand its dimensional formula, consider the elements:

• The dimension of speed, or velocity, is \[L][T]^{-1}\].
• Time's dimension is \[T\].
• Thus, acceleration has the dimension \[(\frac{[L][T]^{-1}}{[T]} = [L][T]^{-2}\].

This dimension shows acceleration as the rate of change of velocity per unit time, reflecting its crucial role in dynamics.

Time

Time is a critical concept for measuring motion. It's a base unit in physics, with the dimension \[T\]. Time allows us to track the sequence and duration of events, playing a fundamental role in analyzing movement.In the context of physics:

• It helps calculate speed \text{(distance/time)}\.
• It measures how long it takes for an object to achieve a certain acceleration.
• Understanding time is essential in experiments and observation.

Time's simplicity as a linear, continuous quantity might disguise its importance, but it is vital for measuring and understanding complex systems of motion.