(a) now take fy--mgf with ky if y< 0, 0 otherwise the new term ff represents the a spring-like force exerted on the mass by a floor at y-0. this force is only present when the height of the mass y(t) goes below zero. take k 104 n/m. now the mass should bounce off the floor; pick a range of t to show several bounces i. show that if the number of time points is too small, the rk2 solution is unstable and diverges to very large numbers (as in the first two problems in this homework set) ii. show that if the number of points is large enough to avoid the instability (e. g. 1000), there is still a significant error in the rk2 solution: the mass bounces to higher and higher heights, violating energy conservation (see newman sec. 8.5.2 for a discussion). the odeint solution does not have this problem iii. show that by using even more time points, the energy viola- tion in the rk2 solution can be made negligible. (b) change the floor force to otherwise where u = dy/de is the velocity of the mass. the new term γν represents a viscous damping force that reduces kinetic energy of the mass during bounces. try γ-10 n-s/m.

Follow these directions and answer the questions. 1. shine a pencil-thin beam of light on a mirror perpendicular to its surface. (if you don't have a laser light as suggested in the video, you can make a narrow beam from a flashlight by making a cone from black construction paper and taping it over the face of the flashlight.) how does the light reflect? how does the relationship of incident to reflected ray relate to the reflection of water waves moving perpendicular to a barrier? 2. shine a pencil-thin beam of light on a mirror standing on a sheet of paper on the table (or floor) so that you can mark the incident ray and reflected ray. (you can support the mirror from the back by taping it to a wooden block.) 3. mark a line on the paper representing the reflective surface. (the reflective surface of a mirror is usually the back edge.) 4. draw a dashed line perpendicular to the mirror surface at a point where the incident and reflected ray meet. this perpendicular is called a normal to the surface. 5. measure the angles between the rays and the normal. the angle of incidence is the angle formed by the incident ray and the normal to the surface. the angle formed by the reflected ray and normal is called the angle of reflection (r). what is the angle of incidence? what is the angle of reflection? 6. repeat for several different angles. (see report sheet for details.) what appears to be the relationship between the angle of incidence and angle of reflection? in science 1204, what was the relationship for these two angles made by the reflection of waves in a ripple tank? 7. roll a ball bearing so that it hits a fixed, hard surface (a metal plate) at several angles (including head-on). observe the way in which the ball bearing reflects. what generalization can you make about how a ball bearing reflects from a wall? have you proved that light can only behave like a wave?

Increasing the pressure on one spot of a fluid will increase the pressure everywhere else in the fluid a. pascals principal b. archimedes' principal c. bernoulli's principal

Which is an example of an exothermic process?