91Ó°ÊÓ

Understanding the concept of power is crucial in this problem. Power measures how quickly work is done. The formula for power is given as \[ P = \frac{W}{T} \]where:

• \( P \) is power,
• \( W \) is the work done,
• \( T \) is the time taken to do the work.

This formula tells us that if power is large, the same amount of work will take less time to complete and vice versa. In the exercise, the relationship is established between two engines, with engine 1 having twice the power of engine 2. Hence, if engine 1 performs work in time \( T \), understanding this formula helps determine how engine 2 with less power requires more time to do the same work.

Time calculation is another critical aspect of understanding work and power. In the exercise, we calculated the time it takes for two different engines to perform the same work. We know that for engine 1\[ P_1 = \frac{W}{T} \]Given that engine 2's power \( P_2 \) is half of that of engine 1, we have:\[ 2P_2 = \frac{W}{T} \]Substituting in the values for power:\[ P_2 = \frac{W}{2T} \]To find the time \( T_2 \) it takes engine 2 to do the work, we use the power formula again:\[ P_2 = \frac{W}{T_2} \]From our substitutions, this simplifies to:\[ \frac{W}{2T} = \frac{W}{T_2} \]Solving the above equation gives \( T_2 = 2T \). This result should make sense because engine 2 has half the power of engine 1, so it takes twice as long to do the same work.

When evaluating different engines, comparing their power allows us to anticipate their performance in performing work over time. The exercise demonstrates a practical example of this by comparing engine 1 and engine 2. Engine 1 has a power denoted by \( P_1 \), approximately twice that of engine 2, \( P_2 \). This means:\[ P_1 = 2P_2 \]What this practically means in the context of the exercise is engine 1 can do work in half the time it takes engine 2 to do the same work. Therefore, engine 2 needs twice the time to complete the same amount of work because of its lower power rating.Knowing this helps in real-world scenarios such as selecting engines for specific tasks, understanding the trade-off between power and work duration, and possibly even managing efficiency and resource allocation in various engineering applications.