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Illustrations of the Thomson and Rutherford models of the atom and the associated experiments. The Thomson model has electrons, illustrated as small solid balls distributed throughout a large, uniform sphere. Alpha particles pass through undeflected. Several trajectories of alpha particles, incident from the left and travelling horizontally to the right are shown as straight, parallel lines that pass through the atom unchanged. The experiment consists of a collimated source of alpha particles. The beam of particles passes through a gap in a screen that surrounds a gold foil target. The beam passes through the target, spreading a little, but hitting the screen in a small spot on the far side of the screen. The expected result is particles detected in only one spot. The Rutherford model has electrons, illustrated as small solid balls distributed throughout the atom, but the nucleus is a small sphere in the center. Several trajectories of alpha particles, incident from the left and travelling horizontally to the right are shown as straight, parallel lines as they enter the atom. Some pass through unchanged, one is bent slightly away from its original direction, and is bent back at an angle large than 90 degrees. The experiment consists of a collimated source of alpha particles. The beam of particles passes through a gap in a screen that surrounds a gold foil target. The beam passes through the target, most of it passing through but spreading significantly and hitting the screen on the far side over an extended region, and a few of the particles hitting the screen on the same side of foil as the source. The expected result is particles detected in many spots.
The Thomson and Rutherford models of the atom. The Thomson model predicted that nearly all of the incident alpha-particles would be scattered and at small angles. Rutherford and Geiger found that nearly none of the alpha particles were scattered, but those few that were deflected did so through very large angles. The results of Rutherford’s experiments were inconsistent with the Thomson model. Rutherford used conservation of momentum and energy to develop a new, and better model of the atom—the nuclear model.

Summary

  • An elastic collision is one that conserves kinetic energy.
  • An inelastic collision does not conserve kinetic energy.
  • Momentum is conserved regardless of whether or not kinetic energy is conserved.
  • Analysis of kinetic energy changes and conservation of momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one-dimensional, two-body collisions.

Conceptual questions

Two objects of equal mass are moving with equal and opposite velocities when they collide. Can all the kinetic energy be lost in the collision?

Yes, all the kinetic energy can be lost if the two masses come to rest due to the collision (i.e., they stick together).

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Describe a system for which momentum is conserved but mechanical energy is not. Now the reverse: Describe a system for which kinetic energy is conserved but momentum is not.

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Problems

A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg bowling pin, which is scattered at an angle to the initial direction of the bowling ball and with a speed of 15.0 m/s.

  1. Calculate the final velocity (magnitude and direction) of the bowling ball.
  2. Is the collision elastic?

a. 6.80 m/s, 5.33°; b. yes (calculate the ratio of the initial and final kinetic energies)

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Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei from gold-197 nuclei. The energy of the incoming helium nucleus was 8.00 × 10 −13 J , and the masses of the helium and gold nuclei were 6.68 × 10 −27 kg and 3.29 × 10 −25 kg , respectively (note that their mass ratio is 4 to 197).

a. If a helium nucleus scatters to an angle of 120 ° during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the gold nucleus.

A helium nucleus (2 protons and 2 neutrons) is incident with velocity v 1 on a gold nucleus. The path of the helium nucleus after the collision makes an angle of 120 degrees from its original direction of travel.

b. What is the final kinetic energy of the helium nucleus?

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A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it takes the puck to reach the goal 15.0 m away?

2.5 cm

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A 100-g firecracker is launched vertically into the air and explodes into two pieces at the peak of its trajectory. If a 72-g piece is projected horizontally to the left at 20 m/s, what is the speed and direction of the other piece?

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In an elastic collision, a 400-kg bumper car collides directly from behind with a second, identical bumper car that is traveling in the same direction. The initial speed of the leading bumper car is 5.60 m/s and that of the trailing car is 6.00 m/s. Assuming that the mass of the drivers is much, much less than that of the bumper cars, what are their final speeds?

the speed of the leading bumper car is 6.00 m/s and that of the trailing bumper car is 5.60 m/s

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Repeat the preceding problem if the mass of the leading bumper car is 30.0% greater than that of the trailing bumper car.

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An alpha particle ( 4 He) undergoes an elastic collision with a stationary uranium nucleus ( 235 U). What percent of the kinetic energy of the alpha particle is transferred to the uranium nucleus? Assume the collision is one-dimensional.

6.6%

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You are standing on a very slippery icy surface and throw a 1-kg football horizontally at a speed of 6.7 m/s. What is your velocity when you release the football? Assume your mass is 65 kg.

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A 35-kg child sleds down a hill and then coasts along the flat section at the bottom, where a second 35-kg child jumps on the sled as it passes by her. If the speed of the sled is 3.5 m/s before the second child jumps on, what is its speed after she jumps on?

1.9 m/s

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A boy sleds down a hill and onto a frictionless ice-covered lake at 10.0 m/s. In the middle of the lake is a 1000-kg boulder. When the sled crashes into the boulder, he is propelled over the boulder and continues sliding over the ice. If the boy’s mass is 40.0 kg and the sled’s mass is 2.50 kg, what is the speed of the sled and the boulder after the collision?

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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