Due November 7, 1995
Reading Assignment: Fourth Day, pages 244, 245, 249-254, First Day pages 1-7, 88-90, Second Day pages 130-132.
1. Straight line motion: sketch qualitatively how downward acceleration varies with time for:
(a) a sky diver, from plane to ground,
(b) a bungee jumper, from start to finish.
For purposes of this question, assume survival of both the above.
2. (WARNING: The only formula allowed in this question is distance = average speed x time taken, which is really just the definition of average speed. Use of any other formulas you may know will be penalized! Just think through the steps as they are given.)
Suppose you throw a ball vertically upwards at 20 meters per second, and assume the natural acceleration due to gravity is 10 meters per second per second downwards.
(a) What is the velocity of the ball after one, two, three and four seconds?
(b) How long did it take the ball to reach its highest point?
(c) What was its average speed on the way up?
(d) Using your results from (b) and (c) figure out how high it got.
(e) Suppose you throw a tennis ball at 45 degrees to the horizontal and it stays in the air four seconds. Ignoring air resistance, how far away from you does it land? (Neglect the height of your hand above the ground.) What was its speed when it left your hand?
(f) Same as (e), except now it stays in the air only two seconds. How far did you throw it? How fast did you throw it?
(g) Find out approximately how far you can throw a tennis ball, and then figure out from your results above about how fast you can throw.
3. (a) Draw a square of side 1 centimeter (this doesn't have to be too precise, but should look close to right). Now draw a square with sides 3 cms. What is the ratio of the areas of the two squares? Prove your answer by showing how a certain number of the small squares would fit in the big one.
(b) How many cubes of side 1 cm. would fit in a cube of side 3cm.? In other words, what is the ratio of the volumes of the two cubes? What is the ratio of the total surface areas of the two cubes? What is the total surface area of the larger cube in square centimeters?
(c) Suppose I have two solid brass cubes, the larger one has 100 times the surface area of the smaller one, and the smaller one weighs one kilogram. How much does the larger one weigh?
(d) Consider a closed curve of any shape (an irregular blob) drawn on a flat piece of paper). Now imagine it magnified, keeping the same shape, so that the distance between the two furthest-apart points on the curve is doubled. By what factor does the area enclosed by the curve increase? (Hint: you could fill up the area inside the curve with a lot of little squares, imagine it drawn on graph paper, using even tinier squares to fill the partially-filled squares around the edges.)
4. In an experiment on scaling carried out in 1883, a dog of mass 3 kg. and surface area 2500 sq. cm. was found to need 300 calories a day to stay alive and warm (no exercise or weight change). Another dog of mass 30 kg. and surface area 10,000 sq. cm. was found to need 1100 calories a day.
(a) Do the calorie requirements correlate better with mass or surface area? Explain what you would expect. How precise would you expect the correlation to be?
(b) Make an estimate of your own mass and surface area, and, assuming you are doglike for purposes of this question, how many calories do you need a day to stay alive, without exercise and at constant weight?
5. The hands of a clock, naturally, go around the dial clockwise, meaning that as a hand passes the topmost point it's moving to the right. Clocks were invented and developed in the northern hemisphere. Explain carefully how these facts might be connected.
6. What did Galileo do or say to get the Jesuit astronomer Scheiner upset? What did he do to get Pope Urban VIII upset? What did Galileo think of Kepler? Was he right?