After understanding work and energy in the previous sections of Chapter 5, the natural next question is: how fast is the work being done? Two machines may perform the same amount of work, but the one that completes it in less time is more powerful. This idea leads us to the concept of Power.
Power connects mechanics to real-world applications such as engines, motors, lifting systems, automobiles, turbines, and even rocket propulsion. In competitive exams like JEE Main and JEE Advanced, power is frequently tested in combination with force, velocity, circular motion, friction, and conservation of energy.
At Deeksha Vedantu, we emphasize conceptual clarity so that students can confidently apply power formulas in both board examinations and high-level competitive problems.
Definition of Power
Power is defined as the rate at which work is done.
Mathematically:
Where:
= power
= work done
= time taken
= work done
= time takenSI unit of power is Watt (W).
This means if 1 joule of work is done in 1 second, the power is 1 watt.
Dimensional Formula of Power
Work has dimension 
.
Therefore power has dimension:
Dimensional understanding is important in advanced problem solving.
Average Power
Average power is defined as total work done divided by total time taken.
This formula is useful when motion is not uniform but total time and total work are known.
Example 1 (Board Level)
A person lifts a 30 kg object to a height of 3 m in 6 seconds.
Work done:
Average power:
This example demonstrates how gravitational work converts into power.
Instantaneous Power
In most real-life situations, force and velocity vary continuously. Hence, we define instantaneous power.
Since infinitesimal work is:
Dividing both sides by 
:
This is one of the most important formulas in mechanics.
Vector Nature of Power
Power depends on the dot product between force and velocity.
Where 
is the angle between force and velocity.
Special Cases
- If
(parallel):
(parallel):
- If
(perpendicular):
(perpendicular):
This explains why centripetal force in uniform circular motion does no work and produces no power.
Power in Terms of Kinetic Energy
From Work–Energy Theorem:
Differentiating with respect to time:
Since:
Thus:
This confirms consistency between definitions.
Units of Power
- SI Unit: Watt (W)
- Horsepower:
Horsepower is commonly used in automobile engines.
Power Delivered by Engines at Constant Speed
When a vehicle moves with constant velocity against resistive force 
:
If speed doubles, required power doubles (assuming resistive force constant).
Example 2 (JEE Type)
A car moves at constant speed 30 m/s against frictional force 500 N.
Power required:
Power While Climbing an Incline
If a body moves up an incline at constant speed:
If friction acts:
Such layered problems are common in JEE Main and Advanced.
Constant Power Engine – Acceleration Analysis
If engine delivers constant power 
:
Using Newton's Second Law:
Thus acceleration:
Acceleration decreases as velocity increases.
Integrating:
Thus velocity increases proportional to square root of time.
This is an important JEE Advanced result.
Power in Circular Motion
In uniform circular motion:
Thus:
Although force exists, no work is done.
However, if tangential force exists (non-uniform circular motion), power is non-zero.
Power and Momentum Relation
Since:
Power can be written as:
This form is useful in advanced dynamics.
Efficiency of Machines
Efficiency measures how effectively input power is converted to useful output power.
Efficiency is always less than 1 due to energy losses.
Example 3 (Elevator Problem)
A 1200 kg elevator moves upward at constant speed 2 m/s.
If efficiency is 75%:
Such multi-step reasoning is common in exams.
Power–Time Graph Interpretation
Total work done:
Area under power–time curve equals work done.
This graphical interpretation is frequently tested.
Advanced JEE Problem 1 (Variable Force)
If force varies with velocity as:
Power:
Power increases rapidly with velocity.
Advanced JEE Problem 2 (Block Pulled by Rope)
A block is pulled with constant power on the horizontal surface.
Since and friction acts, velocity increases gradually.
Acceleration:
Multi-variable reasoning is required.
Advanced JEE Problem 3 (Rocket Insight)
Thrust force produces power:
As rocket speed increases, required power increases.
This explains high fuel consumption at high speeds.
Common Conceptual Mistakes
- Confusing work with power.
- Ignoring angle in dot product.
- Assuming centripetal force does power.
- Forgetting unit conversions.
- Applying average formula in instantaneous cases.
Comprehensive Formula Table
| Concept | Formula |
| Average Power | |
| Instantaneous Power | |
| Force–Velocity | |
| Linear Case | |
| Constant Power Acceleration | |
| Efficiency | |
| Work from Power | |
FAQs
Q1. What is the physical meaning of power?
It represents how fast energy is transferred.
Q2. Why is power zero in circular motion?
Because force is perpendicular to velocity.
Q3. Why does acceleration decrease in constant power motion?
Because force decreases as velocity increases.
Q4. What is horsepower?
Q5. Why is power important in competitive exams?
Because it connects force, energy, motion, and real-world applications.
Conclusion
Power completes the study of work and energy by introducing the rate of energy transfer. The relation provides a powerful bridge between theoretical mechanics and practical systems like engines and motors.
At Deeksha Vedantu, we ensure students gain deep conceptual understanding along with mathematical precision, enabling them to solve even advanced JEE-level power problems confidently.
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