Question

Engine 1 produces twice the power of engine 2 . If it takes engine 1 the time \(T\) to do the work \(W\), how long does it take engine 2 to do the work \(3 W\) ? Explain.

Short answer

Engine 2 takes 6T to do the work 3W.

Step-by-step solution

Understanding the relationship between power, work, and time

In physics, power (P) is defined as the work done (W) divided by the time taken (T). Mathematically, this is represented as \( P = \frac{W}{T} \). As per the problem, Engine 1 is twice as powerful as Engine 2.

Expressing Engine 1's Power

For Engine 1, it takes time \( T \) to do work \( W \). So, the power of Engine 1 is \( P_1 = \frac{W}{T} \).

Expressing Engine 2's Power

Engine 1 produces twice the power of Engine 2, meaning Engine 2's power is half that of Engine 1. So, \( P_2 = \frac{P_1}{2} = \frac{W}{2T} \).

Determining time for Engine 2 to do work 3W

We need to determine the time \( t_2 \) it takes for Engine 2 to do work \( 3W \). Using the formula \( P = \frac{W}{T} \), for Engine 2 doing work \( 3W \), it's \( P_2 = \frac{3W}{t_2} \). Substitute \( P_2 = \frac{W}{2T} \) into this equation: \( \frac{W}{2T} = \frac{3W}{t_2} \).

Solving for \( t_2 \)

Cross-multiply to clear the fractions: \( W \cdot t_2 = 3W \cdot 2T \). Simplify: \( t_2 = 6T \). This means Engine 2 takes 6 times longer to perform the action of 3 times the work compared to Engine 1 doing normal work.

Key concepts

Work-Energy Principle

In physics, the work-energy principle is essential for understanding how energy transformations occur during work done by forces. Work is defined as the product of force and displacement in the direction of the force. Mathematically, it is represented as \( W = F \times d \), where \( F \) is the force applied, and \( d \) is the displacement. Power, on the other hand, is defined as the rate at which work is done. It effectively measures how quickly work is performed or energy is converted. The relationship is given by the formula \( P = \frac{W}{T} \), where \( P \) represents power, \( W \) represents work, and \( T \) represents time.
Understanding this principle helps us to determine how different machines or engines perform over varying periods, as seen with engines in this exercise. If one engine performs work at twice the power of another, it helps us understand the relative efficiency and time it takes to complete specific tasks.

Time Calculation

Calculating the time required for a task is crucial in various physics applications, especially when different powers are involved. In this exercise, we discover how to calculate the time taken by various engines to complete work using their power ratings. The formula for power \( P = \frac{W}{T} \) can be rearranged to find time:\( T = \frac{W}{P} \).
In our exercise, Engine 2 needs to perform three times the work compared to Engine 1. Since we know that Engine 1 takes time \( T \) to do work \( W \), and it is twice as powerful as Engine 2, we can infer the time difference.
Using the information that Engine 1's power \( P_1 \) is twice Engine 2's power \( P_2 = \frac{P_1}{2} \), we substitute and rearrange the equations to find \( t_2 \), resulting in the expression \( t_2 = 6T \). Thus, Engine 2 takes six times longer.

Engine Efficiency

Efficiency in engines relates to how effectively energy is converted into work. It's a measure of energy conservation within an engine when performing tasks. In simple terms, a more efficient engine uses less fuel to perform the same amount of work compared to a less efficient one.
This exercise implicitly touches on efficiency through the power differences between engines. A higher power engine, like Engine 1, accomplishes tasks faster, which can imply greater potential efficiency in energy use. However, true efficiency also considers the input energy, which isn’t part of our present scope, but it is an allied concept that's important for understanding engine performance.

Ratio of Powers

The ratio of powers highlights the relative performance levels between systems, engines, or machines. It refers to comparing two values of power output to understand how they differ in capability.
In the context of our problem, Engine 1 having twice the power of Engine 2 is precisely the key observation we exploit to solve for time. The power ratio informs us that for the same work, Engine 1 completes the task in half the time it would take Engine 2 with the same workload.
The calculation was as follows: if Engine 1’s power is \( P_1 \) and Engine 2’s power is \( P_2 = \frac{P_1}{2} \), Engine 2 takes \( 6T \) to perform \( 3W \) of work because it's doing more work at half the power. This concept is crucial in engineering, where optimizing processes or engine performance depends heavily on understanding and leveraging power ratios.