1.The constant k is given by the formula k = 1/2?CDA where ? is the density of the atmosphere, A is the frontal area of the object, and CD is a dimensionless constant called the “drag coefficient” which measures how aerodynamic the object is. For instance, according to Wikipedia, the box-like Hummer H2 has a drag coefficient of 0.57 and the much more energy-conscious Toyato Prius has a drag coefficient of 0.29. In this question, we will consider a spherical ball, for which we may assume the drag coefficient is CD = 0.47. The frontal area of the ball is A = ?r 2 where r is the radius. We will use ? = 1.225kg/m3 for the density of air.
(i) Calculate the constant k for a bowling ball with a radius of 10.8 centimeters which weighs 7.30 kilograms.
(ii) On the Earth, acceleration due to gravity is given by g = 9.81m/s2 . If a ball as described were dropped from the top of one of the Petronas Towers, a height of 451.9 meters, how long would it take to fall and what would its velocity be upon hitting the ground? Important note: this is a hypothetical question only. We in no way condone the dangerous activity of dropping heavy objects from tall heights.
(iii) On Mars, the force of gravity is weaker, but the air density is much lower meaning it is not immediately clear if objects there will fall slower or faster than on Earth. Suppose a tower of the same height as above was constructed on Mars and a ball dropped from the top of it. How long would the ball take to fall and what would its velocity be upon hitting the Martian ground? Note: to do this question, you will have to go find the relevant data about Mars yourself.
(iv) Using MATLAB or another program, give plots of the vertical distance s(t) of the ball as it falls. The figure should include both the plot for the ball falling on Earth and the ball falling on Mars. Note: if you prefer, you can give a plot of the height h(t) = 451.9 ? s(t) of the ball above the ground as a function of time.