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Chapter 9 Due: 12:00am on Saturday, July 3, 2010
Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy
Introduction to Collisions
Description: Basic questions about two disks colliding, elastically and perfectly inelastically, with varied mass ratios. General definition of elasticity of a collision introduced in the last part. Uses applets.
Learning Goal: To understand how to find the velocities of particles after a collision.
There are two main types of collisions that you will study: elastic and perfectly inelastic. In an elastic collision, kinetic energy is conserved. In a perfectly inelastic collision, the particles stick together and thus have the same velocity after the collision. There is actually a range of collision types, with elastic and perfectly inelastic at the extreme ends. These extreme cases are easier to solve than the in-between cases.
In this problem, we will look at one of these in-between cases after first working through some basic calculations related to elastic and perfectly inelastic collisions.
Let two particles of equal mass collide. Particle 1 has initial velocity , directed to the right, and particle 2 is initially stationary.
Part A If the collision is elastic, what are the final velocities and of particles 1 and 2?
Hint A.1 How to approach the problem In analyzing any collision, you can always use the conservation of momentum as long as there are no external forces acting on the colliding particles. In elastic collisions you can also use the conservation of energy. Each of these conservation laws will allow you to write down an equation relating , , and . Once you have the equations, use algebra to eliminate from the system to get a formula for in terms of . Then, go back to the original system and eliminate to get a formula for in terms of .
Hint A.2 Conservation of momentum Which of the following formulas correctly expresses conservation of momentum for the two particles?
ANSWER:
Since the particles have equal mass, the term will factor out of both sides of the equation, leaving .
Hint A.3 Conservation of energy Which of the following formulas correctly expresses conservation of energy for this part?
ANSWER:
Since the particles have equal mass, (and a factor of ) will factor out of the equation, leaving .
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Give the velocity of particle 1 followed by the velocity of particle 2, separated by a comma. Express each velocity in terms of .
ANSWER: =
Part B Now suppose that the collision is perfectly inelastic. What are the velocities and of the two particles after the collision?
Hint B.1 How to approach the problem In analyzing any collision, you can always use the conservation of momentum. In perfectly inelastic collisions you can also use the fact that the final velocities of the particles are equal. Each of these facts will allow you to write down a linear equation relating , , and . Once you have the equation from conservation of momentum, simply substitute using the equation to find the final velocity of each particle in terms of .
Hint B.2 Conservation of momentum Which of the following formulas correctly expresses conservation of momentum for the two particles?
ANSWER:
Since the particles have equal mass, the term will factor out of both sides of the equation, leaving .
Give the velocity of particle 1 followed by the velocity of particle 2, separated by a comma. Express the velocities in terms of .
ANSWER:
=
Part C Now assume that the mass of particle 1 is , while the mass of particle 2 remains . If the collision is elastic, what are the final velocities and of particles 1 and 2?
Hint C.1 How to approach the problem In analyzing any collision, you can always use the conservation of momentum. In elastic collisions you can also use the conservation of energy. Each of these conservation laws will allow you to write down a linear equation relating , , and . Once you have the equations, you can use algebra to eliminate from the system to get a formula for in terms of . Then, go back to the original system and eliminate to get a formula for in terms of .
Hint C.2 Conservation of momentum Which of the following formulas correctly expresses conservation of momentum for the two particles?
ANSWER:
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Notice that will factor out of both sides of the equation, leaving .
Hint C.3 Conservation of energy Which of the following formulas correctly expresses conservation of energy for this part?
ANSWER:
Notice that will divide out of the equation, leaving .
Give the velocity of particle 1 followed by the velocity of particle 2, separated by a comma. Express the velocities in terms of .
ANSWER:
=
Note that in both the conservation of momentum equation and the conservation of energy equation, cancels out. This is a general feature of many collision situations: The ratio of the two masses is important, but the absolute masses are not.
Part D Let the mass of particle 1 be and the mass of particle 2 be . If the collision is perfectly inelastic, what are the velocities of the two particles after the collision?
Hint D.1 How to approach the problem In analyzing any collision, you can always use the conservation of momentum. In perfectly inelastic collisions you can also use the fact that the final velocities of the particles are equal. Each of these facts will allow you to write down a linear equation relating , , and . Once you have the equation from conservation of momentum, simply substitute using the equation to find the final velocity of each particle in terms of .
Hint D.2 Conservation of momentum Which of the following formulas correctly expresses conservation of momentum for the two particles?
ANSWER:
Notice that will factor out of both sides of the equation, leaving .
Give the velocity of particle 1 followed by the velocity of particle 2, separated by a comma. Express the
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velocities in terms of .
ANSWER:
=
This applet shows two disks colliding. The orange disk has always the same initial velocity. You can change the ratio of the masses of the two disks and the elasticity of the collision. You should try the four different settings corresponding to Parts A through D. An elastic collision has elasticity , and a perfectly inelastic collision has elasticity .
Part E
What qualitative change takes place as the ratio of the mass of the blue disk to the mass of the orange disk, ,
increases from 0.3 to 4.0? Set the elasticity to 1.0 for a perfectly elastic collision.
ANSWER: The final speed of the orange disk decreases as the ratio of masses increases. As the ratio increases past 1.0, the final velocity of the orange disk changes direction. The difference in final velocities between the disks decreases. The difference in final velocities between the disks increases.
Most real collisions are somewhere between elastic and perfectly inelastic. This is indicated by the elasticity of the collision, which measures the difference in the velocities of the particles after the collision compared with the difference in velocities before the collision. For instance, in a perfectly inelastic collision, the two particles stick together after colliding. The elasticity of such a collision is , because the difference in velocities between the particles is 0 after they collide.
Technically, the elasticity is defined by the relation , where and are the initial and final velocities of particle 1, and and are the initial and final velocities of particle 2. In this problem, the formula is simplified by our definition of and the hypothesis . So, using for the final velocity of particle 1 and for the final velocity of particle 2, we obtain the simpler formula .
This final form will be most useful to you in solving Part F.
Part F If the two particles with equal masses collide with elasticity , what are the final velocities of the particles? Assume that particle 1 has initial velocity and particle 2 is initially at rest. Look at the applet to be sure that your answer is reasonable.
Hint F.1 How to approach the problem You know that in any collision you can use the conservation of momentum. This will be the same as it was in Part A for equal masses colliding: . The second equation that you can form is from the definition of elasticity: . Here, . You can solve these two equations to find and .
Give the velocity of particle 1 followed by the velocity of particle 2, separated by a comma. Express the velocities in terms of .
ANSWER: =
Notice that if you look back at your answers to Parts A and C, the diference between and is always , as you would expect from setting in the definition of elasticity. It is possible, though it takes some algebra, to prove that the definition of elasticity with implies conservation of energy.
This applet is the same as the previous one, but now you are given a graph of the momentum for each disk at the bottom. Run a few of the collisions that you have studied in this problem so that you can see how the momenta of the two disks change with differing elasticities and mass ratios.
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Also in this applet you can have the two disks collide off-center. While this looks much more complicated, the law of conservation of momentum still always applies. With a modification to make it more precise for two- and three- dimensional collisions, the definition of elasticity still applies as well.
Problem 9.20
Description: A plutonium-239 nucleus at rest decays into a uranium-235 nucleus by emitting an alpha particle ^4 (He) with kinetic energy of 5.15 MeV. (a) What is the speed of the uranium nucleus? A plutonium-239 nucleus at rest decays into a uranium-235 nucleus by emitting an alpha particle with kinetic energy of 5.15 .
Part A What is the speed of the uranium nucleus?
ANSWER: =
Problem 9.25
Description: An object with kinetic energy K explodes into two pieces, each of which moves with twice the speed of the original object. (a) Compare the internal energy K_int and center-of-mass energy K_cm after the explosion. An object with kinetic energy explodes into two pieces, each of which moves with twice the speed of the original object.
Part A Compare the internal energy and center-of-mass energy after the explosion.
ANSWER: =
Problem 9.23
Description: A m trick baseball is thrown at v. It explodes in flight into two pieces, with a m1 piece continuing straight ahead at v1. (a) How much energy do the pieces gain in the explosion? A 120 trick baseball is thrown at 75 . It explodes in flight into two pieces, with a 50 piece continuing straight ahead at 86 .
Part A How much energy do the pieces gain in the explosion? Express your answer using two significant figures.
ANSWER:
=
Problem 9.45
Description: A ^238 U nucleus is moving in the x direction at v when it decays into an alpha particle ^4 (He) and a ^234 (Th) nucleus. (a) If the alpha particle moves off at phi above the x axis with a speed of v1, what is the speed of the thorium nucleus? (b)...
A nucleus is moving in the direction at 4.7×105 when it decays into an alpha particle and a
nucleus.
Part A
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If the alpha particle moves off at 22 above the axis with a speed of 1.3×107 , what is the speed of the thorium nucleus? Express your answer using two significant figures.
ANSWER:
=
Part B What is the direction of the motion of the thorium nucleus? Express your answer using two significant figures.
ANSWER:
= clockwise from the -axis
Problem 9.72
Description: A m1 projectile is launched at v1 at a phiangle to the horizontal. At the peak of its trajectory it collides with a second projectile moving horizontally, in the opposite direction, at v2. The two stick together and land s horizontally downrange from... A 14 projectile is launched at 380 at a 55 angle to the horizontal. At the peak of its trajectory it collides with a second projectile moving horizontally, in the opposite direction, at 140 . The two stick together and land 9.0 horizontally downrange from the first projectile's launch point.
Part A Find the mass of the second projectile. Express your answer using two significant figures.
ANSWER:
=
Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 44 points.
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