• Question #1:
  "A box is pulled along a level floor at constant speed by a rope that makes a 45 degree angle with the floor. The box weighs 100 N. The coefficient of sliding friction is 0.75. The force exerted on the rope is
  a. 75N
  b. between 75 N and 100 N
  c. 100 N
  d. greater than 100 N"
  
  The book in which this was found says the answer is "d." Do you believe it? Check and see if my answer below agrees with this choice, or any of them! (Anyone think that answer "a" should have said, "less than 75N"?)
• Answer to question #1.

Question #2: The early history of the universe
• The question you asked about deals with the age of the universe. In the fall of 1998, the Astrophysics class looked at the Einstein Field Equations and applied them to a flat-spacetime Robertson-Walker universe, which treats the individual galaxies as an initially homogeneous distribution of "dust particles," or like an "ideal gas." When we run those equations back in time, we get the following graphs for the parameters of that particular model:
  
  1. matter density (I haven't had a chance to figure out how to make the Mathematica graphs look any better than this!)
  2. radiation density
  3. radiation pressure
  4. Hubble parameter
  5. temperature
• The graphs show that all of these parameters--horribly intertangled through the nonlinear differential equations that arise from the field equations--blow up around 12.8 billion years in the past. (The horizontal axis shows time in the past in units of years.)

Terminal velocity
• The question posed: "My question is a 4.5 kg metal sphere is released in a fluid where k=10.5Ns/m. I have to plot a velocity v.s. time graph of it's motion until it reaches 90% of it's terminal velocity and find out how long it took to reach 90% of it's terminal velocity?"
• First, you have to identify the forces on this body and put them into Newton's 2nd Law. The downward force is mg (where g=9.8 m/s^2) while your upward force is kv. Putting these into Newton 2, and choosing down as the positive direction, gives the equation of motion,
  ma = mg - kv. Dividing out the masses gives,
  a = g - (k/m)v. With a=dv/dt (one time derivative of v), you need to integrate,
  dv/dt = g - (k/m)v. (1) Rewriting,
  dv = g dt - (k/m) v dt. Turning differentials into differences gives,
  v_f - v_0 = g (t_f - t_0) - (k/m) v_0 (t_f - t_0) where v_0=the starting velocity at each time step, (t_f - t_0)= your time step (chosen very small), v_f=the ending velocity after each step, and v_0 is chosen to be on the right side since dv=v_f-v_0 deals with the change from this original value. Then, the equation that you need to fill into your spreadsheet column for each new velocity v_f given a starting vecloity v_0 is given by
  v_f = v_0 + g (t_f - t_0) - (k/m) v_0 (t_f - t_0).
  
• I started with an initial downward velocity of 0m/s, and chose a timestep of dt=0.01 sec. I chose a m=1.0kg mass just for kicks; g=9.8 (units m/s^2); and k=10.5 (units Ns/m). After about 0.68 sec, my downward speed hit a speed of around 0.933m/s. My Excel graph of the problem is here.

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