Problem Solving with Energy (Edexcel A Level Further Maths): Revision Note

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, Newtons, when moving metres is  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