Matter & Interactions 3rd ed. Practice Problems
Aaron Titus | High Point University
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46     A comet orbiting Sun     46
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A comet of mass orbits Sun which has a mass . Use the following constants and initial conditions for the comet in your calculations.

Assume that Sun remains stationary during the orbit (i.e. it does not "wobble").

  1. What is the position and momentum of the comet at t = 30 days.
  2. What is the period of the comet's orbit?
  3. What is the position of the comet at aphelion (the point in its orbit when it is furthest from Sun.

 

61     Energy of a muon     61
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A muon is an elementary particle with a negative charge, like an electron, but with a larger mass ( ) than an electron. Suppose that a muon has a constant velocity of magnitude 0.9c, where c is the speed of light.

  1. What is the muon's total energy, in both joules and electron-volts?
  2. What is the muon's kinetic energy, in both joules and electron-volts?

 

62     Work done on a pole vaulter during a vault     62
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The path of a pole vaulter is measured using video analysis. The y vs. x graph (which shows the path) of the pole vaulter is shown in the figure below.


Figure: The path of a pole vaulter.

The mass of the vaulter is 70.0 kg. He plants the pole when he is at the location , and the pole starts to bend. With the pole mostly bent, the vaulter starts his mostly upward trajectory at the location . The average force by the pole on the pole vaulter during this interval is . What is the total work done on the pole vaulter during this interval? State any assumptions that you make.

 

63     Final speed of a space probe with thrusters firing; application of the update form of the energy principle     63
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In a 3-D computer game, a 5000-kg space probe in space (far from any significant interactions with planets or stars) has four thrusters that fire in the +x, –x, +y, and –y directions, respectively. As the space probe moves from to , there are two thrusters simultaneously firing with forces and respectively. If the initial speed of the space probe is 200 m/s, what is its final speed? State any assumptions that you make.


Figure: Thrusters on a spaceship firing in the +y and – x directions, respectively.

 

64     Change in rest energy resulting from the fusion of hydrogen into helium.     64
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In one type of fusion reaction, 6 hydrogen nuclei (each nuclei is one proton) eventually, after many steps, form 1 atom of helium (2 neutrons and 2 protons) and two atoms of hydrogen. (Read more about the proton-proton cycle to learn about the process.) The mass of hydrogen is . The mass of helium is . What is the change in the rest energy of the system during this reaction?

 

65     Kinetic energy of particles resulting from the decay of a neutron.     65
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A free neutron (mass ) is highly unstable, meaning that it won't exist for very long before it decays. When it decays, it decays into a proton (mass , an electron (mass , and an antineutrino. (An antineutrino is a fundamental particle of zero electric charge and zero or very small mass. Particle physicists have not yet determined the mass of the antineutrino. There are actually three types of antineutrinos. You can read about them at Wikipedia at http://en.wikipedia.org/wiki/Standard_Model.

  1. If a neutron at rest decays, what will be the total kinetic energy of the proton, electron, and antineutrino when they are very far apart? An antineutrino has no rest energy (or actually it has negligible rest energy).
  2. Why is it impossible for all three particles to travel in the same direction after the decay?

 

66     Work done on a springs in a piece of exercise equipment     66
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The piece of exercise equipment shown below includes two handles that are on each end of two parallel springs, each with a stiffness 500 N/m. To build strength, a person holds the two handles and pulls them apart with equal magnitude forces applied in opposite directions to the handles, thus stretching the springs. Neglect the mass of the springs and assume that they are pulled horizontally.


Figure: Two parallel springs between two handles that can be pulled apart.

  1. How much work must a person do to stretch the springs 5 cm?
  2. Suppose that a person is already holding the handles so that the springs are stretched 5 cm from their relaxed lengths. How much more work must the person do to stretch the springs an additional 5 cm (i.e. from 5 cm stretched to 10 cm stretched)?

 

67     Initial speed of a space probe traveling from Earth to Jupiter along a ``least energy" orbit     67
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To send a probe from earth to another outer planet, it is most efficient (i.e. conserves the most fuel) to put the probe into an orbit about Sun so that when it is nearest Sun (perihelion), its path is tangent to the earth's orbit, and when it is furthest from Sun (aphelion), its path is tangent to the outer planet's orbit. Besides the fuel necessary to put the probe into orbit (i.e. escape from Earth) and to make it orbit the outer planet once it gets there, no fuel is necessary during travel from Earth to the outer planet. For this reason, the probe's orbit in this case is called a "least-energy" orbit.

Suppose the probe travels from Earth to Jupiter as shown below. The radius of the Earth's nearly circular orbit is , and the radius of Jupiter's nearly circular orbit is .


Figure: The path of a probe leaving Earth at perihelion and arriving at Jupiter at aphelion.

If the probe's speed at aphelion should be the same as the speed of Jupiter ( ), what should be the probe's speed at perihelion, when it leaves Earth with its thrusters turned off? Neglect interactions of the probe with Earth and Jupiter, and assume that the probe's motion is completely determined by its interaction with Sun.

 

68     The work needed to launch the Hubble Space Telescope into orbit.     68
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The Hubble Space Telescope (HST or "the Hubble") has a mass of about 11,100 kg and is in a nearly circular orbit of altitude 560 km.

  1. Use the momentum principle to calculate the speed of the HST.
  2. What is the period of its orbit?
  3. What is the work needed to put the HST into orbit, starting at the surface of Earth?

 

69     Launch speed of rocket from Moon     69
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Moon has a mass of and a radius of . Moon has no atmosphere, so there is no air resistance. Suppose that a manned lunar rocket on the surface of Moon must launch so that when it is very far from Moon, it has a speed of 1500 m/s. What is the required launch speed? (Treat the rocket as a particle with constant mass.)

 


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