Problem Solving with Energy

How do I include air resistance in the Work-Energy Principle?

The work done by a constant air resistance / drag force, $D$ Newtons, when moving $x$ metres is $D x$ Joules
Air resistance hinders (slows down) the particle, so is negative in the Work-Energy Principle
- total final energy = total initial energy - work done by air resistance
This can work for particles moving horizontally or vertically
- sometimes the air resistance experienced upwards has a different value to that experienced downwards
Air resistances, in reality, are often proportional to the speed (or square of the speed) of the particle
- but this makes it a non-constant force
  - and the work done formula only works for constant forces

How do I use the Work-Energy Principle on curved surfaces?

The Work-Energy Principle can be used in new situations that aren't always inclined planes!
e.g. skateboarding down a curving slope
- the skater may put in their own work done (e.g. using their legs) which "helps" to go faster (+ work done)
- but there may be a constant resistive force acting against them throughout (- work done)
  - assume that the resistances are always parallel to the curved slope at any given time (and reactions are always perpendicular)

How do I apply the Work-Energy Principle to connected particles?

You can still use the Work-Energy Principle with connected particles by considering it all as one object
- total final energy = total initial energy ± work done
The total energies will be the sum of the GPEs and KEs of all particles
There will be a combination of "work done" terms with + or - depending on whether it's helping or hindering its respective particle
- e.g. for a driving car pulling a trailer, the terms look like:
- + WD(by driving force on car) - WD(by tension in towbar on car) - WD(by resistances on car) + WD(by tension from towbar on trailer) - WD(resistances on trailer)
  - Notice that the work done by the tensions will cancel each other out

How do I apply the Work-Energy Principle to collisions?

Some questions use the Work-Energy Principle and the theory of collisions
There may be a particle projected into a perpendicular wall
- Use the Work-Energy Principle to find the speed with which it impacts the wall
  - You can find the speed by making the kinetic energy the subject
- This gives the speed of impact
- To find the speed of rebound, calculate "e" × the speed of impact
  - "e" is the coefficient of restitution
Other questions may have two spheres colliding on a horizontal table, then one falling off
- Use conservation of momentum and Newton's Law of Restitution to find velocities after the collision
- When the sphere rolls off the table, it becomes a projectile (projected horizontally with it's new velocity)
  - If you know the height of the table, you can use the Work-Energy Principle to find the speed of impact with the ground

Exam Tip

It is common for harder energy questions to be fully algebraic
- look out for masses, $m$ , cancelling in the working

Worked example

A particle of mass $m$ kg is projected vertically upwards from ground level at a speed of $5 \sqrt{g H}$ ms^-1, where $H$ is the vertical height in metres between the ground and the ceiling. The particle is subjected to a constant air resistance force of $\frac{1}{4} m g$ N, opposing its motion. The coefficient of restitution between the particle and the ceiling is $\frac{\sqrt{2}}{3}$ .

Find, in terms of $g$ and $H$ , the exact speed of the ball immediately after rebounding with the ceiling.

worked example focusing on problem solving aspects of energy questions, utilising algebraic terms

fm1-2-6-2-problem-solving-with-energy-we-solution-2-pf-for-mc

Problem Solving with Energy (Edexcel A Level Further Maths: Further Mechanics 1)

Revision Note