PHY 3221: Mechanics I
Fall Term 2009
Exam 1, September 23 2009
• This is a closed book exam lasting 50 minutes.
• There are two problems worth a total of 20 pts. Begin each problem on a fresh sheet of
paper. Use only one side of the paper. Avoid microscopic handwriting.
• Put your name, the problem number and the page number in the upper right-hand corner
of each sheet.
• To receive partial credit you must explain what you are doing. Carefully labelled figures
are important! Randomly scrawled equations are not helpful.
• Draw a box around important results (or at least results which you think might be important).
• Good luck!
1
Problem 1. Kickoffs in “The Swamp”. [10 pts]
(a) [2 pts] You decide to impress your non-physics friends on campus and claim that without
studying any game film of the Gator football team, you can determine how hard its kickers can
kick a football. Looking up their stats on the internet, you find out that last year the Gators were
averaging L yards on their kickoffs which ended in a fair catch. Using this information alone, determine the speed v0 with which the football leaves the kicker’s foot. Neglect air resistance and
assume that the kicker is always directing the ball at the optimum angle to achieve maximum
distance.
(b) [5 pts] Once word of your physics skills spreads around campus, you are contracted by
the University Athletic Association (UAA) as a consultant on a very special project. In order
to create an unfair advantage for the Gators over the opposition, UAA is considering to install a
mechanism underneath the stadium which can tilt the football field. The tilt can be changed in
between quarters so that the Gators are always attacking downhill (see figure below). If the tilt
angle is β, and the Gator kicker is kicking the ball at an angle α with respect to the horizontal,
show that the range of the Gator kicks is given by
R=
2 v02 cos α sin(α + β)
.
g cos2 β
Once again, ignore the effect of air resistance.
(c) [3 pts] Using the result from part (b), for a given tilt angle β, find the optimal angle α
which would allow for the maximum range Rmax .
~v0
GATORS
α
~g
“THE SWAMP”
OPPONENT
β
Figure 1: An illustration for the Swamp kickoffs problem.
2
Problem 2. Downhill skiing. [10 pts.]
A skier of mass m is standing atop a conical hill of height h and slope θ. The skier then
heads downhill, experiencing along the way a resisting force proportional to the square of the
instantaneous speed: Fr = kmv 2 .
m
~g
h
θ
Figure 2: An illustration for the skier problem.
(a) [8 pts] Find the skier’s speed v at the bottom of the hill.
(b) [2 pts] As a cross-check, discuss as best as you can whether your answer makes sense in
the following extremes: 1) θ → 0; 2) k → ∞; 3) h → ∞.
3
Formula sheet
A·(B × C) = B·(C × A) = C·(A × B) ≡ ABC
A×(B × C) = (A · C)B − (A · B)C
(A × B) · (C × D) = A · [B × (C × D)]
= A · [(B · D)C − (B · C)D]
= (A · C)(B · D) − (A · D)(B · C)
(A × B) × (C × D) = [(A × B) · D] C − [(A × B) · C] D
= (ABD)C − (ABC)D = (ACD)B − (BCD)A
v = ṙ er + r θ̇ eθ + r sin θ φ̇ eφ
a =
r̈ − r θ̇ 2 − r φ̇2 sin2 θ er + 2ṙ θ̇ + r θ̈ − r φ̇2 sin θ cos θ eθ
+ 2ṙ φ̇ sin θ + 2r θ̇φ̇ cos θ + r θ̈ sin θ eφ
v = ṙ er + r φ̇ eφ + ż ez
a =
r̈ − r φ̇2 er + r φ̈ + 2ṙ φ̇ eφ + z̈ ez
X
εijk εlmk = δil δjm − δim δjl
k
X
εijk εljk = 2 δil
X
εijk εijk = 6
j,k
i,j,k
4