Problem Set #1

1.

A rocket ship blasts off at t-0 and accelerates upward at a rate of $5 {\rm m/sec^2}$. A mile away, a news cameraman films the event using a tripod that tilts so that he can keep the rocket in the center of the field of view. What is the rate of change of the camera's angle with the horizon ( $\frac{d \theta}{dt}$)? What are the units of your solution? Justify your answer.

2.

Suppose one wishes to characterize the terminal velocity of a water droplet falling in the atmosphere. The relevant variables would be the characteristic size of the droplet, the acceleration due to gravity, the viscosity of the atmosphere (with units M L^-1T^-1) and the density of the liquid. How many dimensionless combinations of the relevant quanitities are there? Find all dimensionless products of these variables.

3.

A rule of thumb for cooking turkeys is that one should allow 20 minutes of cooking time per pound of turkey in a 400^o oven. Using dimensional analysis, determine whether or not this is a good idea. If you think it is a bad idea, propose a better one.

4.

Read the Scientific American article ``How Dinosaurs Ran'' by R. M. Alexander. Express the principle results of the first half of the article (summarized in the graph on page 132) in terms of Buckingham's $\Pi$ Theorem. Suppose you did not have access to the data in this graph, but you hypothesize that the only relevant variables for quadrupedal animals are speed v, stride length s, hip height l and gravity g. Use whatever resources you can to determine the functional relationship between the dimensionless parameters. (In other words, find animals representing data points with different dimensionless values. Otherwise, your graph will collapse to a point!)