Assume 'A' does 500 J of work in 'x' minutes and 'B' does 1000 J of work in 20 minutes. If the power delivered by 'A' is P1 and 'B' is P2 and P1 = 2P2, then 'x', in minutes is :

1. Understanding Work and Energy (basic)

Welcome! To master mechanics, we must first distinguish between 'effort' and Work. In physics, work is not just about being busy; it is specifically defined as a force acting upon an object to cause a displacement. If you push against a mountain with all your might but it doesn't move, your 'work done' is zero. Scientifically, Work (W) is the product of the force applied and the distance moved in the direction of that force. The standard unit for measuring this is the Joule (J).

While Work tells us what was accomplished, Power (P) tells us how fast it was done. Power is the rate of doing work. If two people lift the same 10kg weight to the same height, they have done the exact same amount of work. However, if the first person does it in 2 seconds and the second takes 10 seconds, the first person is significantly more powerful. Mathematically, this is expressed as P = W/t. This formula reveals an inverse relationship between power and time: for a fixed amount of work, the less time you take, the more power you deliver.

In our daily lives, we often encounter these concepts through electrical appliances or industrial output. For instance, the Watt (W) is the SI unit of power, representing one joule per second. When we talk about large-scale energy consumption, we use the kilowatt-hour (kWh), which is actually a unit of energy representing the work done by a 1000-watt source in one hour, equivalent to 3.6 × 10⁶ Joules Science, Electricity, p.191. Interestingly, the concept of 'work' extends beyond physics into our study of the economy; just as a machine must be functional to have power, an individual in the workforce is considered 'unemployed' only if they possess the ability and willingness to work but cannot find an opportunity Indian Economy, Poverty, Inequality and Unemployment, p.47.

To help you keep these distinctions clear, look at this comparison:

Concept	Core Definition	Standard Unit
Work	Force applied over a distance (F × d)	Joule (J)
Energy	The capacity or 'fuel' to perform work	Joule (J)
Power	The speed or rate of work (W / t)	Watt (W)

Sources: Science, Electricity, p.191; Indian Economy, Poverty, Inequality and Unemployment, p.47

2. Forms of Mechanical Energy (basic)

In our study of basic mechanics, Mechanical Energy is perhaps the most visible form of energy we encounter. It is defined as the energy possessed by an object due to its motion or its position. Unlike thermal energy, which involves the microscopic vibration of molecules Environment and Ecology, Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8, mechanical energy deals with macroscopic objects—like a car moving down a road or a brick held above the ground.

Mechanical energy is the sum of two distinct types of energy:

• Kinetic Energy (KE): This is the energy of motion. Anything that moves has kinetic energy. For instance, the blowing wind possesses kinetic energy which can be captured by the blades of a turbine INDIA PEOPLE AND ECONOMY, NCERT 2025 ed., Mineral and Energy Resources, p.61. The amount of kinetic energy depends on the object's mass and the square of its velocity (KE = ½mv²).
• Potential Energy (PE): This is "stored" energy based on an object's position or arrangement. The most common form is Gravitational Potential Energy, such as water stored behind a high dam (hydel power). Because of its height, the water has the "potential" to fall and do work INDIA PEOPLE AND ECONOMY, NCERT 2025 ed., Mineral and Energy Resources, p.65.

In practical applications, we often see these forms converting into one another. A wind turbine is a perfect example: it captures the kinetic energy of the wind and converts it into mechanical energy by turning a rotor, which then drives a generator to produce electricity Environment, Shankar IAS Academy, Renewable Energy, p.290. Similarly, the food we eat acts as a chemical fuel that our bodies convert into the mechanical energy needed to perform physical work Science, class X, NCERT 2025 ed., Our Environment, p.210.

Feature	Kinetic Energy (KE)	Potential Energy (PE)
Core Concept	Energy of Motion	Energy of Position/State
Example	A flowing river or blowing wind	A stretched rubber band or water in a reservoir
Key Variable	Speed (Velocity)	Height or Displacement

Sources: Environment and Ecology, Majid Hussain (Access publishing 3rd ed.), BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8; INDIA PEOPLE AND ECONOMY, TEXTBOOK IN GEOGRAPHY FOR CLASS XII (NCERT 2025 ed.), Mineral and Energy Resources, p.61, 65; Environment, Shankar IAS Academy (ed 10th), Renewable Energy, p.290; Science, class X (NCERT 2025 ed.), Our Environment, p.210

3. Law of Conservation of Energy (intermediate)

At the heart of classical mechanics lies the Law of Conservation of Energy. This fundamental principle states that energy can neither be created nor destroyed; it can only be transformed from one form to another or transferred from one object to another. In an isolated system, the total amount of energy remains constant. For instance, when we observe planetary movements or electromagnetic radiation, we are seeing energy shifting between kinetic, potential, and radiative forms without any net loss to the universe. Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267.

In the natural world, this principle is visible in the flow of energy through ecosystems. Primary producers like plants capture solar energy and, through photosynthesis, convert it into chemical energy (carbohydrates). This energy is then passed to consumers. However, it is crucial to understand that while total energy is conserved, its quality or usability changes. According to the laws of thermodynamics, whenever energy is transformed to do work, a portion is dissipated—usually as heat—making it unavailable for further work. This is why energy flow in the biosphere is described as unidirectional rather than cyclic. Environment and Ecology by Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14.

From a policy perspective, the term "energy conservation" (as seen in India's Energy Conservation Act, 2001) refers to reducing the waste of energy. Since energy resources like coal are exhaustible and their use involves transforming energy into less useful forms, efficiency is key. By using cleaner fuels and better technology, we maximize the work extracted from energy before it inevitably dissipates. Contemporary World Politics, Environment and Natural Resources, p.90. This transition from fossil fuels to renewables is a strategic application of managing energy transformations to ensure sustainable development. Geography of India by Majid Husain, Energy Resources, p.8.

Sources: Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267; Environment and Ecology by Majid Hussain, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.14; Contemporary World Politics, Environment and Natural Resources, p.90; Geography of India by Majid Husain, Energy Resources, p.8

4. Introduction to Power (basic)

In our previous discussions, we looked at Work as the product of force and displacement. However, in the real world, it isn’t just about how much work is done, but how fast it is accomplished. This rate of doing work is what we define as Power. Think of it this way: if two administrative officers are tasked with digitizing 1,000 files, and Officer A finishes in two days while Officer B takes four, both have performed the same amount of work, but Officer A has shown greater "power" in their execution.

Mathematically, Power (P) is the ratio of Work (W) to the Time (t) taken to complete it: P = W / t. Because work is also a measure of energy transfer, power can similarly be defined as the rate at which energy is consumed or dissipated Science, Class X (NCERT 2025 ed.), Electricity, p.191. The SI unit of power is the Watt (W), named after James Watt. One Watt is defined as the power of an agent which does work at the rate of 1 Joule per second (1 W = 1 J/s).

In practical and industrial applications, the Watt is often too small a unit. We frequently use Kilowatts (kW), where 1 kW = 1000 Watts Science, Class X (NCERT 2025 ed.), Electricity, p.191. It is crucial to distinguish between power and energy: power is a rate, while energy is the total capacity. This is why our electricity bills use the Kilowatt-hour (kWh); this is a unit of energy (Power × Time), representing the energy consumed when 1000 Watts are used for one hour, equivalent to 3.6 × 10⁶ Joules Science, Class X (NCERT 2025 ed.), Electricity, p.192.

Concept	Definition	Formula / Unit
Work	Energy transferred by a force	Joule (J)
Power	The speed/rate at which work is done	Watt (W) = J/s

Sources: Science, Class X (NCERT 2025 ed.), Electricity, p.191; Science, Class X (NCERT 2025 ed.), Electricity, p.192

5. Mechanical Efficiency and Simple Machines (intermediate)

In our journey through mechanics, we must understand that while Work is the transfer of energy, we often use tools called Simple Machines to make that transfer easier. Whether it is a pulley used in a rural well or a thresher in a field, the goal is to gain a Mechanical Advantage—this means using a smaller input force to move a larger load. However, there is a fundamental rule in physics: you never get something for nothing. In any simple machine, if you decrease the amount of force required, you must increase the distance over which that force is applied. The total work done remains the same (or slightly more) because energy is conserved.

This leads us to the concept of Mechanical Efficiency. Efficiency is defined as the ratio of Useful Work Output to the Total Work Input, usually expressed as a percentage. In an ideal world, efficiency would be 100%, but in reality, some energy is always lost to the environment—primarily as heat due to friction. For instance, when mechanical power is supplied to tube-wells or crushers Geography of India, Majid Husain, Agriculture, p.49, the moving parts experience resistance. Therefore, the actual work performed by the machine is always less than the work put into it. Improving efficiency is a core challenge in engineering, especially in power plants where resource conservation is critical Environment and Ecology, Majid Husain, Distribution of World Natural Resources, p.22.

To calculate efficiency, we use the formula: Efficiency (η) = (Work Output / Work Input) × 100. It is important to distinguish efficiency from Power. While power is the rate at which work is done (P = W/t), efficiency tells us how well the energy is being used. A high-power engine might get a job done quickly, but if it has low efficiency, it will waste a significant amount of fuel in the process. This distinction is vital for sustainable development and resource management.

Concept	Focus	Key Relationship
Power	Speed of work	Inverse to time (Work / Time)
Efficiency	Quality of energy use	Output vs. Input (Output / Input)

Sources: Geography of India, Agriculture, p.49; Environment and Ecology, Distribution of World Natural Resources, p.22

6. Mathematical Relationships in Power (intermediate)

In our previous discussions, we established that work is the transfer of energy. However, in both physics and administration, it isn't just about what is done, but how fast it is done. This brings us to Power (P), which is mathematically defined as the rate of doing work. Just as speed is the distance covered in a unit of time Science-Class VII, Measurement of Time and Motion, p.113, power is the work done divided by the time taken: P = W/t. This formula reveals a vital inverse relationship: for a fixed amount of work, if you increase the power, the time required to complete the task decreases. Conversely, if you take more time to do the same work, you are operating at a lower power. To master these relationships, think of power as a measure of intensity or productivity. In economics, we often look at how output changes when we vary inputs like labor or time Microeconomics (NCERT class XII 2025 ed.), Production and Costs, p.39. Similarly, in mechanics, if we compare two machines (or people), we can determine their relative power by looking at their output (work) over a specific duration. If Machine A does 500 Joules of work in 5 minutes, its power is 100 J/min. If Machine B does 1000 Joules in 20 minutes, its power is 50 J/min. Even though Machine B did more total work, Machine A is twice as powerful because its rate is higher. Understanding these ratios allows us to solve for any missing variable. For instance, if you know the power capacity of a system and the total work required, you can calculate the time using t = W/P. This is similar to how we calculate the time difference between longitudes based on the Earth's constant rate of rotation Certificate Physical and Human Geography, The Earth's Crust, p.11. The mathematical consistency across these disciplines—whether it's the speed of a pendulum Science-Class VII, Measurement of Time and Motion, p.118 or the delivery of energy—is what makes 'rate' such a foundational concept for a civil servant to grasp.

Relationship	Mathematical Form	Conceptual Meaning
Power & Work	P ∝ W (Time constant)	More work in the same time requires more power.
Power & Time	P ∝ 1/t (Work constant)	Doing the same work faster requires more power.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113, 118; Microeconomics (NCERT class XII 2025 ed.), Production and Costs, p.39; Certificate Physical and Human Geography , GC Leong (Oxford University press 3rd ed.), The Earth's Crust, p.11

7. Solving the Original PYQ (exam-level)

This question perfectly illustrates the fundamental relationship between Power, Work, and Time. As you recently studied in the Rate of Doing Work module, Power ($P$) is defined by the formula $P = W/t$. To solve this, you must apply this building block sequentially: first, determine the baseline power of 'B', and then use the comparative ratio provided ($P_1 = 2P_2$) to solve for the missing variable. This multi-step application is a classic way the UPSC tests your ability to handle proportions and algebraic rearrangement.

Let's walk through the coach's logic. Start with 'B': $1000 \text{ J} / 20 \text{ min} = 50 \text{ J/min}$. The problem states 'A' is twice as powerful ($P_1 = 2 \times 50 = 100 \text{ J/min}$). Now, look at 'A's work: $500 \text{ J}$. To find the time $x$, we rearrange the formula to $t = W/P$. Thus, $500 \text{ J} / 100 \text{ J/min} = \mathbf{5}$ minutes. This calculation highlights the inverse relationship between power and time; because 'A' is more powerful and has less work to do, the time must be significantly shorter than 'B's time.

Regarding the other options, UPSC often includes "trap" numbers that result from incomplete processing. Option (A) 10 is a common mistake where a student might see that 'A' does half the work ($500 \text{ J}$ vs $1000 \text{ J}$) and incorrectly assume the time is also halved ($20 / 2 = 10$), ignoring the power ratio entirely. Option (C) 20 assumes the times are identical despite the difference in power. By methodically calculating each step rather than guessing based on visual patterns, you arrive at the correct answer: (B) 5.