MAT 699 Mathematical Modeling Assignment 3 Due Tuesday, October 2
1. Read the attached pages from I.B. Cohen’s The Birth of a New Physics and respond to the following. a. State Kepler’s Three Laws of Planetary Motion.
b. Kepler was in many ways both modern and medieval. Give an example from Kepler’s life and accomplishments that reflects modern science and given an example that reflects medieval modes of thought.
1 Problems
1 2 These problems involve using Galileo’s law of free fall d= at or more generally 2 1 2 d= v t + at and v= at . 0 2
1. An object is dropped from rest and falls freely.
a. After 6 seconds how far has it fallen (in meters)? Use 9.8 m/s per second as the acceleration due to gravity.
b. After 6 seconds, what is its downward speed (in m/s)? Use 9.8 m/s per second as the acceleration due to gravity.
2. a. You throw a ball straight upward at a speed of 20 m/s, how long (in seconds) will it take reach zero speed? Use 10 m/s per second as the acceleration due to the gravity.
b. You throw a ball straight upward at a speed of 20 m/s, how long (in seconds) will it take to return to it starting point? Use 10 m/s per second as the acceleration due to the gravity.
3. a. You throw a ball straight upward at a speed of 20 m/s, how fast will it be going (in m/s) when it returns to its starting point? Use 10 m/s per second as the acceleration due to the gravity.
2 b. Find the speed in m/s required to throw a ball straight up and have it return 6 seconds later. Use 10 m/s per second as the acceleration due to the gravity.
4. Calculate the hang time of an athlete who jumps a vertical distance of 0.75 m. Use 9.8 m/s per second as the acceleration due to gravity and round your answer to the nearest one-hundredth of a second. (HINT: You will need to multiply by 2…)
5. If a ball is thrown straight upward, it height in feet is given by the formula 2 -16t + v0 t where v0 is the initial velocity (in feet per second) and where t is the time after it is thrown (in seconds). If you would like the ball to reach a maximum
height of 100ft, what is the initial velocity v0 with which you have to throw it? Express your final answer both in feet per second and also in miles per hour.
3 6. If you throw a ball straight up in the air, the height of the ball in feet is given by 2 h( t )= - 16 t + v0 t where v is the initial speed of the ball when it leaves your hand. Suppose you throw a ball a ball twice, the second time with an initial speed that is three times as large as the first. How much higher does the ball go the second time compared to the first throw? Show your work.
7. A man wishes to determine the water level in a deep well. He drops a stone into the well and hears the water 3 seconds later. If the stone drops 16t 2 after t seconds, and the speed of sound is 1090 ft/sec, how far below the opening of the well is the surface of the water?
4 8. As I mentioned in class, Galileo also discovered the equation for the shape of the trajectory of a projectile.
Let me review the ideas. If the initial speed is v 0 , the horizontal coordinate of the object 1 2 is x= vcos(θ) t . The vertical coordinate is y= vsin(θ) t - at . If we eliminate t by 0 0 2 solving for t using the first equation and plugging into the second equation, we get
骣 a 2 y=tan(θ) x - 琪 2 2 x 桫2v0 cos (θ) which is, of course, a parabola.
骣 a 2 a. Derive the equation y=tan(θ) x - 琪 2 2 x . 桫2v0 cos (θ)
5 b. Find R, the range of the projectile using the formula
骣 a 2 y=tan(θ) x - 琪 2 2 x . (Hint. This is the other x value where y = 0.) 桫2v0 cos (θ)
c. Find the angle θ that maximizes the range R. (Hint. While one can do this without Calculus, I would be inclined to use Calculus. If you use the relation sin(2θ)= 2sin(θ)cos(θ) , it is quite easy.)
6 9. Extra Credit. An athlete stands at the peak of a hill that slopes downward uniformly at angle f . At what angle from the horizontal should she throw a rock so that it has the greatest range? (Hints. The rock hits the ground at the intersection of two curves. If f = 60º, then θ = 15º.)
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