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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

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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

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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

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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]

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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

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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

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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

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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

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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

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