Overview (Leaving Cert Applied Maths): Revision Notes

Overview

Work, energy, and power are fundamental concepts in physics that describe how forces cause changes in motion and position. Understanding these relationships is essential for solving mechanics problems and forms the foundation for many advanced physics topics.

Work

Work is the energy transferred when a force acts through a distance. It represents the effort needed to move an object against a resistance force.

• Definition: Work is calculated as the product of force and the distance moved in the direction of that force
• Formula: $W = F \times d$
• Units: Joules (J)
• Key point: Work is only done when there is movement in the direction of the applied force

When you push a heavy box across the floor, you're doing work because you apply force and the box moves through a distance. If you push against a wall that doesn't move, no work is done despite applying force.

Power

Power measures how quickly work is done or energy is transferred. It tells us the rate at which energy changes hands.

• Definition: Power is the rate of doing work or transferring energy
• Two equivalent formulas:
  • $P = F \times v$ (Force × Velocity)
  • $P = \frac{W}{t}$ (Work ÷ Time)
• Units: Watts (W)

The first formula shows that power depends on both the force applied and the speed of movement. The second formula reveals that doing the same amount of work in less time requires more power. A car engine that can accelerate quickly has high power output.

Impulse

Impulse describes the effect of a force acting over time and relates directly to changes in momentum.

• Definition: Impulse equals the change in momentum of an object
• Formula: $I = mv - mu$ (where v is final velocity, u is initial velocity)
• Units: Newton seconds (Ns)

Energy types

Kinetic energy

Kinetic energy is the energy an object possesses due to its motion.

• Formula: $KE = \frac{1}{2}mv^2$
• Units: Joules (J)
• Key insight: Energy increases with the square of velocity, so doubling speed quadruples kinetic energy

Potential energy

Potential energy is stored energy based on an object's position, particularly its height above a reference level.

• Formula: $PE = mgh$ (where g is gravitational acceleration, h is height)
• Units: Joules (J)
• Application: Water stored behind a dam has gravitational potential energy

Conservation laws

Conservation of energy

The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another.

Real-world example: A pendulum continuously converts between kinetic and potential energy as it swings back and forth.

Conservation of momentum

The law of conservation of momentum applies when objects interact without external forces.

Application: Used to analyse car crashes, rocket propulsion, and sports collisions.

Drag forces

Drag forces oppose motion through fluids like air or water. Understanding these forces is crucial for analysing real-world motion.

• Mathematical model: $D = kv^n$
• Key variables: k is a constant depending on object shape and fluid properties, v is velocity, n is typically 1 or 2
• Terminal velocity: At maximum speed, acceleration equals zero because drag force balances the driving force

This explains why objects falling through air eventually reach a constant terminal velocity rather than accelerating indefinitely.