Understanding Momentum and Energy: An Introduction to Particle Kinematics and Newton's Laws

1. Handout II : Momentum &Energy 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 http://ppewww.ph.gla.ac.uk/~parkes/teaching/PK/PK.html October 2004

2. Projectiles Motion of a thrown / fired object mass m under gravity Velocity components: vx=v cos  vy=v sin  y Force: -mg in y direction acceleration: -g in y direction v x,y,t  x x direction y direction a: v=u+at: s=ut+0.5at2: ax=0 ay=-g vx=vcos  + axt = vcos  vy=vsin  - gt x=(vcos )t y= vtsin  -0.5gt2 This describes the motion, now we can use it to solve problems

3. Linear Momentum Conservation • Define momentum p=mv • Newton’s 2nd law actually • So, with no external forces, momentum is conserved. • e.g. two body collision on frictionless surface in 1D before m1 m2 v0 0 ms-1 Initial momentum: m1 v0 = m1v1+ m2v2 : final momentum after m1 m2 v2 v1 For 2D remember momentum is a VECTOR, must apply conservation, separately for x and y velocity components

4. Energy Conservation • Energy can neither be created nor destroyed • Energy can be converted from one form to another • Need to consider all possible forms of energy in a system e.g: • Kinetic energy (1/2 mv2) • Potential energy (gravitational mgh, electrostatic) • Electromagnetic energy • Work done on the system • Heat (1st law of thermodynamics of Lord Kelvin) • Friction  Heat Energy measured in Joules [J]

5. m1 m2 v2 v1 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.: ½m1 v02= ½ m1v12+ ½ m2v22 : final k.e. • m1>m2 • m1<m2 • m1=m2 See lecture example for cases of elastic solution Newton’s cradle

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

7. Work & Energy Work is the change in energy that results from applying a force F s • 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 F  x mgh of water

8. This stored energy has the potential to do work Potential Energy We are dealing with changes in energy h • choose an arbitrary 0, and look at  p.e. 0 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

9. Work done by a variable force Consider small distance dx over which force is constant F(x) Work W=Fx dx So, total work is sum dx X 0 F Graph of F vs x, integral is area under graph work done = area dx For spring,F(x)=-kx: X x F X Stretched spring stores P.E. ½kX2