1. USSC2001 Energy Lecture 2 Kinetic Energy in Motion Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 1 Email matwml@nus.edu.sg R:\public_html\courses\Undergraduate\USC\2008\USC2001// Tel (65) 6516-2749

2. NEWTON’s 2 nd LAW The net force on a body is equal to the product of the body’s mass and the acceleration of the body. Question: what constant horizontal force must be applied to make the object below (sliding on a frictionless surface) stop in 2 seconds? 2

3. FALLING BODIES Consider a particle thrown upward with velocity from the ground at time t = 0 Question : What is the particles height h as a function of t and when does the particle hit the ground ? 3

4. HEIGHT COMPUTATION From Newton’s 2 nd Law, the velocity v of the particle as a function of t is given by 4 therefore and the particle hits the ground at

5. ENERGETIC STUPIDITY Fundamentals of Physics by D. Halliday, R. Resnick and J. Walker, p. 117 : "In 1896 in Waco Texas William Crush of the 'Katy' railway parked two locomotives at opposite ends of a 6.4 km long track, fired them up, tied their throttles open, and allowed them to crash head on in front of 30,000 spectators. Hundreds of people were hurt by flying debris; several were killed. Assume that each locomotive weighed 1.2 million Newtons and that its acceleration along the track was equal to Question : What was the total kinetic energy of the two locomotives just before the collision? 5

6. ENERGY COMPUTATION Answer The velocity and distance of each train (in the direction that it is accelerating) satisfies Since each train had mass 6 therefore(a valuable formula) Therefore the squared velocity of each train upon impact was the total energy of the 2 trains was This is the explosive energy of a 44Kg TNT bomb !

7. WORK-KINETIC ENERGY THEOREM Consider a net force that is applied to an object having mass m that is moving along the x-axis The work done is Newton’s 2 nd Law Chain Rule Kinetic Energy 7

8. CONSERVATION OF ENERGY Definition and in that case we can also compute the work as is a potential energy function if so the total energyis conserved since 8

9. HARMONIC OSCILLATOR Conservation of energy often provides a method to derive the equations of motion of physical systems. angular frequency phase period amplitude Energy Conservation  E is constant  9 displacement 

10. VECTORS AND THEIR GEOMETRY Nonnegative Numbers, Real Numbers, Vectors represent quantities having Magnitude, Magnitude+Sign, Magnitude+Direction same vector + = geometric vector addition u u vu w multiplication by a scalar 1.5u -.6u scalar product a b v 10

11. VECTORS AND THEIR ALGEBRA Vectors can be represented by their coordinates and so can vector operations 11

12. NEWTON’s 3rd LAW & MOMENTUM CONSERVATION When two bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction. The momentum of a body is the product of its mass and velocity, the momentum of a system is the sum of the momenta of its bodies. The momentum of an isolated system is conserved. For two bodies this is derived by applying Newton’s 2 nd and 3 rd Laws 12

13. COLLISIONS ALONG A LINE Since when two objects collide the system momentum then A collision is elastic if the kinetic energy is conserved. is conserved 13

14. Inertial Systems and Galilean Transformations Definition 2 physical reference frames are (mutually) inertial if they move with constant relative velocities. This is the case if and only if they admit coordinate systems and that are related by Definition: Galilean Transformation Equations (GTE) 14 Problem: use GTE to show that the velocities seen in two frames satisfy

15. TUTORIAL 2 1. (from Halliday, Resnick and Walker, p.162) A 60kg skier starts from rest at a height of 20 m above the end of ski-jump ramp as shown below. As the skier leaves the ramp, his velocity makes an angle of 28 degrees with the horizontal. Ignoring friction and air resistance, use conservation of energy to compute the maximum height h of his jump? end of ramp 15 Neglect the effects of air resistance and assume that the ramp is frictionless.

16. TUTORIAL 2 2. Show that for a pendulum with small L then use conservation of energy to derive the equations of motion for the pendulum. 3. Derive the two equations on the bottom of page 13. 4. Derive the Principle of Galilean Invariance for elastic collision: kinetic energy is conserved as seen in an inertial reference: Hint: in reference with vel. = 16