Question

Analyze Engine 1 produces twice the power of engine 2 . Is it correct to conclude that engine 1 does twice as much work as engine 2? Explain.

Short answer

No, Engine 1 only does twice the work if both operate for the same amount of time.

Step-by-step solution

Understand Power and Work

Power is the rate at which work is done, or energy is transferred. The relationship between power, work, and time can be defined by the equation \( P = \frac{W}{t} \), where \( P \) is power, \( W \) is work, and \( t \) is time. To compare work done by two engines, we need to consider both power and the time each engine operates.

Analyze Given Information

According to the problem, Engine 1 produces twice the power of Engine 2, which can be expressed as \( P_1 = 2 \times P_2 \). This indicates that Engine 1 can perform work at a faster rate compared to Engine 2. However, this does not directly tell us about the total amount of work done by each engine without considering the time they operate.

Evaluate Work Done by Each Engine

The work done by each engine is determined by \( W_1 = P_1 \times t_1 \) and \( W_2 = P_2 \times t_2 \). To compare their work, we need to assess either the operational time or the total work output. If Engine 1 and Engine 2 operate for the same time (\( t_1 = t_2 \)), then \( W_1 = 2 \times W_2 \). Otherwise, different operational times could result in different amounts of total work.

Conclusion Based on Time Consideration

Without specific information about the operational times \( t_1 \) and \( t_2 \) of the engines, we cannot conclusively state that Engine 1 does twice the work. Therefore, Engine 1 doing twice the work depends on both having equal operational times, \( t_1 = t_2 \). If the times are different, the total work done by each engine will vary accordingly.

Key concepts

Energy Transfer

In physics, energy transfer is a key concept that involves the process of moving energy from one place or form to another. This can occur in numerous ways, such as through mechanical work, heat, or electricity. Understanding energy transfer is essential when analyzing how engines work, as engines convert different types of energy (like chemical energy in fuel) into mechanical energy that can perform tasks.
The rate at which energy is transferred is crucial, as it determines how efficient and powerful a machine can be. When we say "Engine 1 produces twice the power of Engine 2," it means Engine 1 transfers energy at a faster rate, allowing it to perform tasks more quickly if the operational time is the same.
It's important to distinguish between energy transfer and the total energy used or work done over a specific period. The efficiency and capabilities of an engine can be determined by examining how well it uses the energy it transfers during its operation.

Work Done by Engines

Engines do work by converting one form of energy into mechanical energy, which then allows them to move an object or generate force over a specific distance. The work done by an engine is influenced by its power and the time for which it operates.
Work, in physics, is quantified by the equation \[ W = F \times d \] where \( W \) represents work, \( F \) is the force applied, and \( d \) denotes the distance over which the force is applied. For engines, work done can also be calculated in terms of the power output over a specified operational time, given as \[ W = P \times t \].
If Engine 1 has twice the power of Engine 2 and both engines operate for the same time, Engine 1 will indeed do twice the work of Engine 2. However, if the operational times differ, then the total work done will also depend on these times.

Rate of Work

The rate of work, or power, is an indication of how quickly work can be done or how fast energy is transferred. Power is defined as the amount of work done per unit of time and is expressed by the equation \[ P = \frac{W}{t} \], where \( P \) is power, \( W \) is work, and \( t \) is time.
Higher power means quicker work completion or energy transfer. When comparing two engines, like in the original exercise, having twice the power does not inherently mean double the work. It implies that Engine 1 can transfer energy or perform work twice as fast compared to Engine 2 during the same operational period.
This highlights why knowing the operational time is critical—without it, we cannot determine if one engine truly achieves double the work merely through power assessment.

Operational Time

Operational time refers to the duration an engine or machine runs to perform work. It is an essential factor in determining the total work output.
In the equation \[ W = P \times t \], the operational time \( t \) is crucial, alongside power \( P \), to ascertain the work \( W \). Even if one engine has a higher power output, differences in how long each engine operates can alter the final work done.
For instance, if Engine 1 and Engine 2 operate for different times, Engine 1 might not do twice the work of Engine 2 despite having double the power. Only when their operational times are equal will Engine 1 unequivocally perform twice the work as Engine 2, assuming its power is indeed double. Therefore, understanding operational time is vital to making accurate comparisons between engines or machines.