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Solution for (a)

The first ball bounces directly into the wall and exerts a force on it in the + x size 12{+x} {} direction. Therefore the wall exerts a force on the ball in the x size 12{ - x} {} direction. The second ball continues with the same momentum component in the y size 12{y} {} direction, but reverses its x size 12{x} {} -component of momentum, as seen by sketching a diagram of the angles involved and keeping in mind the proportionality between velocity and momentum.

These changes mean the change in momentum for both balls is in the x size 12{ - x} {} direction, so the force of the wall on each ball is along the x size 12{ - x} {} direction.

Strategy for (b)

Calculate the change in momentum for each ball, which is equal to the impulse imparted to the ball.

Solution for (b)

Let u size 12{u} {} be the speed of each ball before and after collision with the wall, and m size 12{m} {} the mass of each ball. Choose the x size 12{x} {} -axis and y size 12{y} {} -axis as previously described, and consider the change in momentum of the first ball which strikes perpendicular to the wall.

p xi = mu ; p yi = 0 size 12{p rSub { size 8{"xi"} } = ital "mu""; "p rSub { size 8{"yi"} } =0} {}
p xf = mu ; p yf = 0 size 12{p rSub { size 8{"xf"} } = - ital "mu""; "p rSub { size 8{"yf"} } =0} {}

Impulse is the change in momentum vector. Therefore the x -component of impulse is equal to 2 mu and the y size 12{y} {} -component of impulse is equal to zero.

Now consider the change in momentum of the second ball.

p xi = mu cos 30º ; p yi = –mu sin 30º size 12{p rSub { size 8{"xi"} } = ital "mu""cos 30"°"; "p rSub { size 8{"yi"} } = - ital "mu""sin 30"°} {}
p xf = mu cos 30º ; p yf = mu sin 30º size 12{p rSub { size 8{"xf"} } = - ital "mu""cos 30"°"; "p rSub { size 8{"yf"} } = - ital "mu""sin 30"°} {}

It should be noted here that while p x size 12{p rSub { size 8{x} } } {} changes sign after the collision, p y size 12{p rSub { size 8{y} } } {} does not. Therefore the x size 12{x} {} -component of impulse is equal to 2 mu cos 30º size 12{ - 2 ital "mu""cos""30"°} {} and the y size 12{y} {} -component of impulse is equal to zero.

The ratio of the magnitudes of the impulse imparted to the balls is

2 mu 2 mu cos 30º = 2 3 = 1 . 155 . size 12{ { {2 ital "mu"} over {2 ital "mu""cos""30" rSup { size 8{ circ } } } } = { {2} over { sqrt {3} } } =1 "." "155"} {}

Discussion

The direction of impulse and force is the same as in the case of (a); it is normal to the wall and along the negative x size 12{x} {} - direction. Making use of Newton’s third law, the force on the wall due to each ball is normal to the wall along the positive x size 12{x} {} -direction.

Our definition of impulse includes an assumption that the force is constant over the time interval Δ t size 12{Δt} {} . Forces are usually not constant . Forces vary considerably even during the brief time intervals considered. It is, however, possible to find an average effective force F eff that produces the same result as the corresponding time-varying force. [link] shows a graph of what an actual force looks like as a function of time for a ball bouncing off the floor. The area under the curve has units of momentum and is equal to the impulse or change in momentum between times t 1 and t 2 size 12{t rSub { size 8{2} } } {} . That area is equal to the area inside the rectangle bounded by F eff , t 1 , and t 2 . Thus the impulses and their effects are the same for both the actual and effective forces.

Figure is a graph of force, F, versus time, t. Two curves, F actual and F effective, are drawn. F actual is drawn between t sub1 and t sub 2 and it resembles a bell-shaped curve that peaks mid-way between t sub 1 and t sub 2. F effective is a line parallel to the x axis drawn at about fifty five percent of the maximum value of F actual and it extends up to t sub 2.
A graph of force versus time with time along the x size 12{x} {} -axis and force along the y size 12{y} {} -axis for an actual force and an equivalent effective force. The areas under the two curves are equal.

Making connections: take-home investigation—hand movement and impulse

Try catching a ball while “giving” with the ball, pulling your hands toward your body. Then, try catching a ball while keeping your hands still. Hit water in a tub with your full palm. After the water has settled, hit the water again by diving your hand with your fingers first into the water. (Your full palm represents a swimmer doing a belly flop and your diving hand represents a swimmer doing a dive.) Explain what happens in each case and why. Which orientations would you advise people to avoid and why?

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
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s. Reply
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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what is the Synthesis, properties,and applications of carbon nano chemistry
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
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or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, Une: physics for the health professions. OpenStax CNX. Aug 20, 2014 Download for free at http://legacy.cnx.org/content/col11697/1.1
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