A) Determine the Speed of the Projectile As a Function of T

PHYS 301/MATH 355

HOMEWORK #10

Due: Monday, April 2, 2007

1. Consider a projectile of mass m launched vertically upward with an initial speed vo. Atmospheric friction is proportional to the speed with a proportionality constant k. Use the appropriate equations of motion to answer the questions below:

a) Determine the speed of the projectile as a function of t.

b) Determine the position of the projectile as a function of time.

c) Determine the time to reach the highest point of ascent.

d) Determine the altitude of this highest point.

e) Show that the results in a)-c) reduce to the familiar results from the zero friction case as k -> 0. (For these questions, think carefully about how many terms in the power series expansion you need to retain.)

[10 points each part]

2. Consider a projectile of mass m launched with a speed of vo at a launch angle θ. If atmospheric friction is proportional to speed with a proportionality constant k, use the appropriate equations of motion to determine:

a) the horizontal speed of the projectile as a function of time

b) the vertical speed of the projectile as a function of time. (can you use your answers from question 1 in any way?)

c) the range (horizontal distance traveled) of the projectile.

[10 pts each part]

3. A train of mass m travels with constant speed v0 along a horizontal track. a) How long will it take for the locomotive to come to rest after the ignition is turned off if the resistance to the motion is given by α + βv2 where α and β are constants? What is the distance traveled? [15 pts total]

4. A particle moves along the x axis acted upon only by a resistive force which is proportional to the cube of the instantaneous speed. If the initial speed is v0 and after a time τ the speed is ½ v0, what will be the speed at time 5τ? (You will need to use both conditions to solve this problem.) [10 points]