Question

Use the two rule-of-thumb approaches (by mass and by cost) to estimate the energy investment in a farm tractor, whose mass is \(2,400 \mathrm{~kg}\) and cost is \(\$ 25,000\). If the results are even within a factor-of-two of each other, we can conclude that the estimate is probably pretty decent as a rough guide.

Short answer

Answer: No, the factor of difference between the two estimates (9.6) is greater than two, which means that they are not within a decent range as a rough guide.

Step-by-step solution

Estimate Energy Investment by Mass

To estimate the energy investment by mass, we can use the rule of thumb that states the energy investment is about \(50 \,\mathrm{~MJ / kg}\). So, for a tractor with a mass of \(2,400 \,\mathrm{~kg}\), we can calculate the energy investment as follows: Energy in MJ = mass of tractor (kg) x 50 MJ/kg Energy in MJ = \(2,400 \, \mathrm{~kg} \times 50 \,\mathrm{~MJ / kg} = 120,000 \, \mathrm{~MJ}\) The energy investment by mass is approximately 120,000 MJ.

Estimate Energy Investment by Cost

To estimate the energy investment by cost, we can use the rule of thumb that states the energy investment is about \(0.5 \,\mathrm{~MJ / \$}\). So, for a tractor with a cost of \(25,000\), we can calculate the energy investment as follows: Energy in MJ = cost of tractor (\() x 0.5 MJ/\) Energy in MJ = \(\$25,000 \times 0.5 \,\mathrm{~MJ / \$} = 12,500 \,\mathrm{~MJ}\) The energy investment by cost is approximately 12,500 MJ.

Compare the Two Estimates

Now, let's compare the two estimates to check if they are within a factor of two of each other: - Energy investment by mass: 120,000 MJ - Energy investment by cost: 12,500 MJ Factor of difference = \(\frac{\mathrm{Energy\, investment \,by\, mass}}{\mathrm{Energy\, investment \,by\, cost}} = \frac{120,000 \,\mathrm{~MJ}}{12,500 \,\mathrm{~MJ}} ≈ 9.6\) Since the factor of difference is greater than 2, we cannot conclude that the estimates are decent as a rough guide.

Key concepts

Mass-Energy Relationship

The concept of the mass-energy relationship signifies a straightforward approach to estimating energy requirements based purely on the mass of an object. In everyday scenarios, this can be highly useful, especially in determining energy investments in large agricultural or industrial equipment.
According to the rule of thumb often used in physics, each kilogram of an object demands approximately 50 megajoules (MJ) of energy investment. This estimation allows us to translate mass into energy terms without complex calculations.

• For example, a farm tractor weighing 2,400 kg translates into an energy investment of 120,000 MJ. This is computed by multiplying the mass of the tractor by 50 MJ/kg.

Despite its simplicity, this relationship provides a helpful, albeit rough, measure of the energy footprint of heavy machinery. However, it should be noted that various factors such as material composition and production processes can affect the actual energy needs, making this rule more of a guideline than a definitive measure.

Cost-Energy Analysis

Analyzing the energy investment based on cost offers another rule of thumb that simplifies estimations for energy requirements. Cost-energy analysis attempts to determine energy expenditures by looking at the monetary value of an item.
This guide is based on the assumption that each dollar spent corresponds to an energy investment of approximately 0.5 MJ. Thus, by multiplying the cost of an asset by this factor, we can gauge the expected energy consumption.

• For instance, considering a farm tractor priced at $25,000, the energy investment amounts to 12,500 MJ. This is calculated using the relationship of cost multiplied by 0.5 MJ/$.

Just like estimating by mass, cost-energy analysis serves as a general framework to quickly gauge energy needs. However, since different items entail varying manufacturing processes and energy efficiencies, results may widely vary, and it is prudent to use this approach alongside other methods for more accuracy.

Rule of Thumb in Physics

A rule of thumb in physics is a practical, easily memorized guideline that gives reasonable estimates, often used in situations where exact measurements are challenging. These rules are excellent for making quick decisions but should be applied cautiously.
In energy estimations, rules of thumb provide rapid insights without delving into complex calculations. They work exceptionally well for approximations like the energy assessment of machinery via mass-energy or cost-energy calculations.

• Despite their expedience, respect for their limitations is crucial. For example, for our tractor, while the energy estimate by mass revealed 120,000 MJ, the estimate by cost was only 12,500 MJ, illustrating a considerable discrepancy.

Therefore, the results remind us that while rules of thumb serve invaluable purposes in quick evaluations, they should not replace detailed analyses, especially when precision is required. Using multiple estimates and averaging them can sometimes lead to more reliable approximations.