MATH 315: Fall 2024

Assignment 1

Due: Monday, September 16

Quote of the Day

The idea that something – anything – could be known with certainty... was delightful, intoxicating, especially when ... it opened up the possibility that other things too might be amenable to strict, mathematical proof. Perhaps even disputes between people might be resolved in this way. ‘I hoped,’ Russell wrote in Portraits from Memory, ‘that in time there would be a mathematics of human behavior as precise as the mathematics of machines.’

- Ray Monk, Bertrand Russell The Spirit of Solitude

I. Mathematical Models.

Read Preface and Chapter 1 of our text by this Wednesday.

Read the excerpts from David R. Causton, A Biologist's Mathematics, London: Edward Arnold, 1977.

watch?v=H7de1sTeD6w

Write up solutions for Exercises 10, 12 - 16, 26, and 27 in Chapter 1 and the problem on Vertical Motion with a Retarding Force on the next page. Although this problem has many parts, each one only involves material from single variable calculus and some algebraic manipulations. The link Solving y’ = ay+ b on our course website may be useful.

For Exercise 10, you may wish to consider some of your favorite similes or metaphors. Here are a couple of famous ones:

“A dream deferred dries up like a raisin in the sun” - Langston Hughes

“The righteous shall flourish like the palm tree,

and grow mighty like a cedar in Lebanon.

Planted in the house of the Lord,

They shall flourish in the courts of our God.

Even in old age they shall bring forth fruit,

They shall be full of vigor and strength.”

- Psalm 92

You may also wish to comment on the confusion caused by such clumsy “mixed metaphors” as Milking the migrant workers for all they were worth, the supervisors barked orders at them.

Vertical Motion With a Retarding Force

Suppose we project a ball of mass m vertically upward from the Earth’s surface with a positive initial velocity . Let y(t) represent the height of the ball above the surface at time t while v(t) represents its velocity.

Assume that the gravitational force g is constant and that there is a retarding force (e.g., friction, air resistance) opposite to the direction of motion and having magnitude proportional to the velocity.

1) Show that in both the ascent and descent of the ball, the total force acting on the ball at time t is

2) Explain why Newton’s Second Law of Motion implies

3) Solve the differential equation and show

4) Integrate the expression for v(t) to find y(t) as an explicit function of t.

5) Let t* be the time it takes the ball to ascend to its maximum height where its velocity is 0. Show that

6) Explain why implies that the time of descent exceeds the time of ascent; that is, the ball spends more time falling than rising.

7) Find as an explicit function of t.

8) Let and show

9) Explain why we know x > 1.

10) Using your result from (7), show

11) Consider the function for x ≥ 1. Sketch a graph of F and use its derivative to give a convincing argument that F is a strictly increasing function.

12) Explain why is positive and hence the time of ascent is less than the time of descent.

13) (Optional Extra Credit): The assumption that the retarding force is proportional to velocity is a reasonable one in some situations, but fails in others. The drag on a skydiver or parachutist is better approximated as proportional to the square of the velocity, for example. The drag on a golf ball (because of spin) is more accurately assumed to be proportional to
Suppose all we know about the retarding force – f(v) is that it is a continuously differentiable function of velocity that satisfies

that is, f(v) > 0 if v > 0 and f(v) < 0 if v < 0.
Show that the time of ascent is less than the time of descent.