EE1 Particle Kinematics : Newton’s Legacy "I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me." Chris Parkes October 2004 Handout II : Momentum &Energy

Projectiles Force: -mg in y direction acceleration: -g in y direction Motion of a thrown / fired object mass m under gravity y x x,y,t v  Velocity components: v x =v cos  v y =v sin  x direction y direction a: v=u+at: s=ut+0.5at 2 : a x =0 a y =-g v x =vcos  + a x t = vcos  v y =vsin  - gt This describes the motion, now we can use it to solve problems x=(vcos  )ty= vtsin  -0.5gt 2

Linear Momentum Conservation Define momentum p=mv Newton’s 2 nd law actually So, with no external forces, momentum is conserved. e.g. two body collision on frictionless surface in 1D before after m1m1 m2m2 m1m1 m2m2 v0v0 0 ms -1 v1v1 v2v2 For 2D remember momentum is a VECTOR, must apply conservation, separately for x and y velocity components Initial momentum: m 1 v 0 = m 1 v 1 + m 2 v 2 : final momentum

Energy Conservation Need to consider all possible forms of energy in a system e.g: –Kinetic energy (1/2 mv 2 ) –Potential energy (gravitational mgh, electrostatic) –Electromagnetic energy –Work done on the system –Heat (1 st law of thermodynamics of Lord Kelvin) Friction  Heat Energy can neither be created nor destroyed Energy can be converted from one form to another Energy measured in Joules [J]

Collision revisited We identify two types of collisions –Elastic: momentum and kinetic energy conserved –Inelastic: momentum is conserved, kinetic energy is not Kinetic energy is transformed into other forms of energy Initial k.e.: ½m 1 v 0 2 = ½ m 1 v ½ m 2 v 2 2 : final k.e. m1m1 v1v1 m2m2 v2v2 See lecture example for cases of elastic solution 1.m 1 >m 2 2.m 1 <m 2 3.m 1 =m 2 Newton’s cradle

Efficiency Not all energy is used to do useful work e.g. Heat losses (random motion k.e. of molecules) –Efficiency  = useful energy produced total energy used ×100% e.g. coal fired power station Boiler Turbine Generator electricity steam coal  40% Chemical energyheatSteam,mechanical workelectricity Oil or gas, energy more direct :  70% Product of efficiencies at each stage

Work & Energy Work = Force F ×Distance s, units of Joules[J] –More precisely W=F.x –F,x Vectors so W=F x cos  e.g. raise a 10kg weight 2m F=mg=10*9.8 N, W=Fx=98*2=196 Nm=196J The rate of doing work is the Power [Js-1  Watts] Energy can be converted into work –Electrical, chemical –Or letting the weight fall –(gravitational) Hydro-electric power station Work is the change in energy that results from applying a force F s x F  mgh of water

This stored energy has the potential to do work Potential Energy We are dealing with changes in energy 0 h choose an arbitrary 0, and look at  p.e. This was gravitational p.e., another example : Stored energy in a Spring Do work on a spring to compress it or expand it Hooke’s law BUT, Force depends on extension x Work done by a variable force

Consider small distance dx over which force is constant F(x) dx Work W=F x dx So, total work is sum 0 X Graph of F vs x, integral is area under graph work done = area F X dx For spring,F(x)=-kx: F x X Stretched spring stores P.E. ½kX 2