Question

Show that the equation \(v_{\mathrm{f}}=v_{\mathrm{i}}+a t\) is dimensionally consistent. In this equation, \(v_{\mathrm{f}}\) and \(v_{\mathrm{i}}\) are velocities, \(a\) is an acceleration, and \(t\) is time.

Short answer

The equation is dimensionally consistent as both sides have the dimension \([L][T]^{-1}\).

Step-by-step solution

Identify Units of Each Quantity

Identify the units of each quantity in the equation. Velocity, both initial \(v_{\mathrm{i}}\) and final \(v_{\mathrm{f}}\), is measured in meters per second \((\mathrm{m/s})\). Acceleration \(a\) is measured in meters per second squared \((\mathrm{m/s^2})\). Time \(t\) is measured in seconds \((\mathrm{s})\).

Analyze Dimensions of Each Term

Break down each term of the equation into its fundamental dimensions. For velocities \(v_{\mathrm{i}}\) and \(v_{\mathrm{f}}\), the dimension is \([L][T]^{-1}\). For acceleration \(a\), the dimension is \([L][T]^{-2}\), and time \(t\) has the dimension \([T]\). Thus, the product \(a \cdot t\) has the dimensions \([L][T]^{-2}][T] = [L][T]^{-1}\).

Compare Both Sides of the Equation

Ensure that the dimensions on both sides of the equation are identical. The left side, \(v_{\mathrm{f}}\), has the dimension \([L][T]^{-1}\). The right side is \(v_{\mathrm{i}} + at\). Since \(v_{\mathrm{i}}\) is simply velocity, it has the dimension \([L][T]^{-1}\). We previously found that \(at\) also has the dimension \([L][T]^{-1}\), so the sum \(v_{\mathrm{i}} + at\) remains consistent with \([L][T]^{-1}\).

Conclusion of Dimensional Consistency

Both sides of the equation \(v_{\mathrm{f}} = v_{\mathrm{i}} + at\) have the same dimensions \([L][T]^{-1}\). This confirms that the equation is dimensionally consistent.

Key concepts

Velocity

Velocity is all about how fast something is going in a specific direction. It's a little different from speed, which just tells you how fast something is moving. Think of velocity as speed with a direction. If you're driving north at 50 kilometers per hour, for example, that's your velocity.

In science and physics, velocity is expressed in meters per second (\( \text{m/s} \)). This unit tells us two things: how far something goes and how quickly it gets there. Understanding this concept is crucial, especially when dealing with equations that describe motion, such as \( v_{\mathrm{f}} = v_{\mathrm{i}} + a t \). The \( v_{\mathrm{i}} \) and \( v_{\mathrm{f}} \) in the equation represent initial and final velocities, respectively.

• Velocity includes both magnitude and direction.
• Measured in meters per second (\( \text{m/s} \)).
• Crucial for studying how objects move.

That's why keeping track of direction is important when calculating or understanding velocity. In problems involving motion, the initial velocity, \( v_{\mathrm{i}} \), is often known, and understanding it helps predict where an object will be in the future.

Acceleration

Acceleration is how much the velocity of an object changes over time. Imagine you're pressing the gas pedal in a car; the car speeds up, so its velocity increases. That change in velocity is acceleration. If you slow down, that's negative acceleration or deceleration.

In terms of physics, acceleration is measured in meters per second squared (\( \text{m/s}^2 \)). This unit means you're looking at how much velocity changes every second, for each second that passes. The concept of acceleration becomes really handy when you're trying to understand equations like \( v_{\mathrm{f}} = v_{\mathrm{i}} + a t \). Here, \( a \) stands for acceleration, influencing the change from initial to final velocity.

• Acceleration measures the change in velocity.
• Expressed as meters per second squared (\( \text{m/s}^2 \)).
• Can be positive (speeding up) or negative (slowing down).

When acceleration is zero, the object maintains its velocity. That's why understanding acceleration is key, so you can see why velocities change. This concept helps you predict motion accurately, which is fundamental in solving motion-related problems.

Time

Time is that ever-present part of our universe that measures the duration of events and the intervals between them. In physics, especially when examining motion, time is crucial because it helps us understand when events occur.

Time is typically measured in seconds (\( \text{s} \)). When analyzing equations like \( v_{\mathrm{f}} = v_{\mathrm{i}} + a t \), time \( t \) is the interval over which acceleration affects velocity. It's like a stopwatch that tracks how long something speeds up or slows down.

• Time keeps track of duration.
• Measured in seconds (\( \text{s} \)).
• Essential for understanding motion and changes in velocity.

Understanding time helps us see how long it takes for changes in an object's motion to happen. Whether calculating the total time of travel or just the interval over which speed changes, time reveals the full picture. This makes it a vital component in equations that describe motion.