Problem 4.2: A ball is dropped from a height of 10 m onto a hard floor. We make the entirely unreasonable assumptions that the motion of the ball is in the vertical direction only. While moving through air we assume that the ball is subject to two forces only, gravity and drag, i.e. air friction. In doing so we ignore a number of other forces such as buoyancy. Justify this assumption numerically. 'We model the retarding force due to air friction by the standard formula: 2 F; =1 pCyAv where pis the air density, C is the drag coefficient, 4 is the area transverse to the direction of motion and v is the speed, i.e. the magnitude of the velocity. Look up a sensible value for the air density. The drag coefficient of a sphere is about 0.47. The ball in question is a table tennis ball. It is not perfectly smooth and will in general have some spin, this raises the drag coefficient to about 0.5. It has a mass of 2.7 g and a diameter of 40 mm. Confirm that these values are reasonable for a competition table tennis/ping pong ball. Obviously the interactions which take place between the ball and the ground are highly complex. Some of the kinetic energy of the ball is lost in each bounce, being converted into acoustic energy, thermal energy, vibrations in the ground, etc. There exists a concept called the coefficient of restitution which purports to model this interaction in a very elementary, highly simplistic manner. Newtonian physics argues for the conservation of linear momentum. Accordingly, the ball actually causes the earth to move at a very small speed. Obviously we choose to ignore this. As a result, although it is defined in a slightly more sophisticated manner, in this case the coefficient of restitution simply equals the ratio of the upward speed immediately after the bounce to the downward speed immediately before it. In effect this method of modelling the complex mechanics of the bounce itself is to not model them, but rather make a bland statement concerning their effect. The actual interaction involved in the bounce also takes some time. We will make the assumption that this interaction time is the same with each bounce being 1 msec. This is not a reasonable assumption. Although it is commeon to assume the coefficient of restitution to be a constant this is not, in general, true. Experimentation suggests that the coefficient of restitution depends upon the relative speed at impact being lower for higher speeds. We will take the following simple formula for the coefficient of restitution: P M