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Homework #3
Due: Friday, Feb 7 Read Chapter 3 Please pass in solutions to 7 problems; 6 of the 7 problems are written
below, and 1 of the 7 problems is in this week’s Supplemental Homework in
"MasteringPhysics". Each problem will be weighted equally. Note: Except for one problem, this week’s MasteringPhysics assignment is
still for practice (weights will be 0points). There are many problems accompa—
nying Chapter 3, so you may not have time to try them all. A number of these
are "tutorials", which may be helpful. 1. Recently, there has been considerable discussion about how to alleviate
concussions occurring among football players, particularly professionals who
continue to play for many years. There is mounting evidence that repeated
concussions have longterm consequences, such as early—onset dementia and de—
pression. Football helmet manufacturers have responded by proposing new de
signs to soften the blow. The key parameter determining the extent of the brain
damage in collisions is the displacement d of the helmet padding as a player’s
head decelerates from a top speed of (typically) 10 m/s, to zero, in a head—on
collision. Larger helmets have room for thicker and squishier padding, which
allows for larger (1. I Using a model of constant acceleration (in this case constant deceleration)
for simplicity, what is the deceleration experienced by a player’s head for a
typical value of d = 2.0 cm? Express your answer in "g’s" (multiples of 9.8
m/s2.) How will this change if the padding is made thicker and softer, so that
for the same collision d = 4 cm? What is the maximum thickness of the padding
in a helmet above which players will look so silly wearing them that people will
no longer tunein to watch football? 2. The maximum range for a projectile launched at an angle 90 with speed
v0 from a ﬂat surface, in the absence of air friction, is given by 2 .
R = 110 $13200) (1) This expression can be derived by combining the equation giving the horizontal
displacement versus time, :12 = mg + newt + 5amt2, with the equation giving the vertical displacement versus time, 1 2
y=y0+voyt+§ayt. You know that am = O, cry 2 —g, because gravity only affects the motion in the
y—direction. You bring the launch angle into the picture by resolving the launch
velocity 60 into its a: and y components , 00$ = 710 cos 60 and my = no sin 60. You
then have to ﬁgure out how long it takes for the object to return to the ground by
setting y = 0 and solving for t. Substituting this time into the equation for a: will
tell you how far the object travels in the a: — direction before it hits the ground.
To put your result into the form above, you should make use of the trig. identity
for the sine of the sum of two angles, sin (0: + ﬂ) : sina cos ﬂ + cos oz sin [3, and
let both angles a and ,8 be equal to 60. Assuming that you can follow these steps, your assignment is as follows: (a) Derive eq. (1) while carefully following the instructions above. (b) Repeat the derivation without looking at the instructions, using a clean
sheet of paper. (0) Keep repeating the derivation until you can do it in less than 5 minutes! None of your work for steps (a—c) needs to be passed in. But now that you
are up to speed, here is something that you should do, to be passed in as the ‘ answer to this problem: (d) Derive what would be the modiﬁcation of the range formula (1) for the ‘ case where the projectile is launched from the top of a building of height h. 3. In the 1991 World Track and Field Championships in Tokyo, Mike Powell
jumped 8.95 m, breaking the 23—year old longjump record set by Bob Beamon
in Mexico City. Assume that Powell’s speed on takeoff was 9.5 m/s, and that
g = 9.80 m/s2 in Tokyo. How much less was Powell’s range than the maximum
possible range for a particle launched at the same speed? 4. A football can be kicked at an initial speed of 25 m/s. What are the (a)
least and (b) greatest elevation angles at which the football can be kicked to score a ﬁeld goal from a point 50 m' in front of goalposts whose horizontal bar
is 3.44 m above the ground? 5. A ball is thrown horizontally from a height of 20 m, and hits the ground
with a speed that is three times its initial speed. (a) What is its initial speed? (b) What is its angle of impact with the ground? (Take straight down to be 90
degrees.) 6. For women’s volleyball, the top of the net is 2.24 In above the ﬂoor, and
the court. measures 9.0 m by 9.0 m on each side of the net. Using a jump serve,
a player strikes the ball at a point that is 3.0 In above the ﬂoor and a horizontal
distance of 8.0 m from the net. If the initial velocity of the ball is horizontal,
(a) what minimum magnitude must it have if the ball is to clear the net, and
(b) what maximum magnitude can it have if the ball is to strike the floor inside
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