1. USSC2001 Energy Lecture 3 Relative Motion Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 6874-2749

2. FALLING OBJECT SEEN FROM THE GROUND Lecture 1 described the motion of a falling object as seen from the ground. The height h is a function of t. The velocity is also a function of time.

3. FALLING OBJECT SEEN FROM AN ELEVATOR How does the motion appear from an elevator that is moving with a uniform speed w ? Question: What is the height of the object seen by an observer who measures it relative to the elevator floor ? Answer:

4. FALLING OBJECT SEEN FROM AN ELEVATOR How does the motion appear from an elevator that is moving with a uniform speed w ? Question: What is the height of the elevator floor as a function of time? Answer:

5. FALLING OBJECT SEEN FROM AN ELEVATOR How does the motion appear from an elevator that is moving with a uniform speed w ? Question: What is the velocity of the object relative to the elevator ? Answer:

6. FALLING OBJECT SEEN FROM AN ELEVATOR How does the motion appear from an elevator that is moving with a uniform speed w ? Question: What is the potential energy of the object relative to the elevator ? Answer:

7. FALLING OBJECT SEEN FROM AN ELEVATOR How does the motion appear from an elevator that is moving with a uniform speed w ? Question: What is the kinetic energy of the object relative to the elevator ? Answer:

8. FALLING OBJECT SEEN FROM AN ELEVATOR How does the motion appear from an elevator that is moving with a uniform speed w ? Question: What is the total energy of the object relative to the elevator ? Answer:

9. TUTORIAL 3 2. Conservation of total energy with respect to an observer on the ground does not imply that an object moves since a stationary object does not change its energy. Prove that if the total energy is constant for every observer moving with constant velocity then the object’s moves according to 1.If a falling object attains maximal height above the Earth’s surface at time T when will it appear to reach a maximal altitude with respect to an elevator that moves with constant velocity w ?

10. COLLISIONS A collision is an isolated event in which two or more bodies exert relatively strong forces on each other for a relatively short time Collisions range widely in scale. Meteor Crater in Arizona (1200m wide by 200m deep), alpha particle bouncing off a nitrogen nucleus, tennis ball in contact with a racquet for 0.004s

11. ELASTIC COLLISIONS We will restrict our attention to systems that are closed (no mass leaves or enters) and isolated (no net external forces act on the bodies within the system) A collision between two bodies is elastic if their total kinetic energy is conserved (same before and after the collision). Although real world collisions are inelastic, many are approximately elastic. Completely inelastic collisions occur when the colliding bodies stick to each other and kinetic energy is transferred into heat.

12. CONSERVATION OF ENERGY RELATIVE TO ALL OBSERVERS Consider the 1-dim elastic collision of two objects Since kinetic energy is conserved relative to an observer moving with velocity w

13. CONSERVATION OF MOMENTUM Consider the 1-dim elastic collision of two objects Question What is momentum and why is it preserved? Answer Momentum is mass times velocity. Expanding both sides of the preceding equation yields an equality between two quadratic polynomials in the variable, the coefficients of the linear terms are equal therefore

14. ANALYTIC GEOMETRY PICTURE The ellipse and line intersect in two points that describe the velocity pairs before and after collision, these points have coordinates

15. TUTORIAL 3 3. Use the preceding two equations to show that and 4. Solve these equations to show that 5. Show that two hockey pucks with identical mass colliding elastically along a line exchange velocities.