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Check Your Understanding There is a second solution to the system of equations solved in this example (because the energy equation is quadratic): v 1,f = −2.5 m/s , v 2,f = 0 . This solution is unacceptable on physical grounds; what’s wrong with it?

This solution represents the case in which no interaction takes place: the first puck misses the second puck and continues on with a velocity of 2.5 m/s to the left. This case offers no meaningful physical insights.

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Thor vs. iron man

The 2012 movie “The Avengers” has a scene where Iron Man and Thor fight. At the beginning of the fight, Thor throws his hammer at Iron Man, hitting him and throwing him slightly up into the air and against a small tree, which breaks. From the video, Iron Man is standing still when the hammer hits him. The distance between Thor and Iron Man is approximately 10 m, and the hammer takes about 1 s to reach Iron Man after Thor releases it. The tree is about 2 m behind Iron Man, which he hits in about 0.75 s. Also from the video, Iron Man’s trajectory to the tree is very close to horizontal. Assuming Iron Man’s total mass is 200 kg:

  1. Estimate the mass of Thor’s hammer
  2. Estimate how much kinetic energy was lost in this collision

Strategy

After the collision, Thor’s hammer is in contact with Iron Man for the entire time, so this is a perfectly inelastic collision. Thus, with the correct choice of a closed system, we expect momentum is conserved, but not kinetic energy. We use the given numbers to estimate the initial momentum, the initial kinetic energy, and the final kinetic energy. Because this is a one-dimensional problem, we can go directly to the scalar form of the equations.

Solution

  1. First, we posit conservation of momentum. For that, we need a closed system. The choice here is the system (hammer + Iron Man), from the time of collision to the moment just before Iron Man and the hammer hit the tree. Let:
    • M H = mass of the hammer
    • M I = mass of Iron Man
    • v H = velocity of the hammer before hitting Iron Man
    • v = combined velocity of Iron Man + hammer after the collision

    Again, Iron Man’s initial velocity was zero. Conservation of momentum here reads:
    M H v H = ( M H + M I ) v .

    We are asked to find the mass of the hammer, so we have
    M H v H = M H v + M I v M H ( v H v ) = M I v M H = M I v v H v = ( 200 kg ) ( 2 m 0.75 s ) 10 m s ( 2 m 0.75 s ) = 73 kg.

    Considering the uncertainties in our estimates, this should be expressed with just one significant figure; thus, M H = 7 × 10 1 kg .
  2. The initial kinetic energy of the system, like the initial momentum, is all in the hammer:
    K i = 1 2 M H v H 2 = 1 2 ( 70 kg ) ( 10 m/s ) 2 = 3500 J.

    After the collision,
    K f = 1 2 ( M H + M I ) v 2 = 1 2 ( 70 kg + 200 kg ) ( 2.67 m/s ) 2 = 960 J.

    Thus, there was a loss of 3500 J 960 J = 2540 J .

Significance

From other scenes in the movie, Thor apparently can control the hammer’s velocity with his mind. It is possible, therefore, that he mentally causes the hammer to maintain its initial velocity of 10 m/s while Iron Man is being driven backward toward the tree. If so, this would represent an external force on our system, so it would not be closed. Thor’s mental control of his hammer is beyond the scope of this book, however.

Practice Key Terms 4

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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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