Galilean Transforms-Relativity

  • Thread starter psingh
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In summary: In this case, because the total momentum of the two fragments is still conserved, the lighter fragment will go faster than the heavier fragment. The lighter fragment has a velocity of 110 m/sec relative to the ground, while the heavier fragment has a velocity of 90 m/sec.
  • #1
psingh
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Homework Statement


A 9.0 kg artillery shell is moving to the right at 100 m/s when suddenly it explodes into two fragments, one twice as heavy as the other. Measurements reveal that 900 J of energy are released in the explosion and that the heavier fragment was in front of the lighter fragment. Find the velocity of each fragment relative to the ground.


Homework Equations



I already have the solution 80 m/s for the heavier and 110 m/s for the lighter but i just want to understand why because i got it wrong on my quiz.


The Attempt at a Solution



i used the elastic collision galilean transforms, but came up with the wrong number.

9kg=x+2x so the heaver is 6kg and the lighter is 3 kg

then i plugged into

mu1'final=(m1-m2)/(m1+m2)*mu1'inital

and

mu2'final=2m/(m1+m2)*mu1'inital

but these were incorrect using 100 m/s as mu1'initial

any help?
 
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  • #2
A change of this sort in the flying shell is not an elastic collision, but is referred to as a "superelastic collision", because the total amount of mechanical energy has increased, due to release from an internal energy source (in this case, the chemical explosion; but releasing a compressed spring initially connected to both masses would have a similar effect).

Conservation of linear momentum still applies, but you must now also keep track of the kinetic energy change, as you would in a partially inelastic collision. Keep in mind that the center of mass still moves in the same direction at 100 m/sec before and after the explosion. Consider the "center-of-mass frame" (hence the Galilean relativity reference): before the explosion, the total KE is zero; afterwards, the total KE of the two fragments is 900 J. You know that the two fragments have masses of 3 and 6 kg. You must now solve two equations simultaneously, the one for which the total linear momentum of the fragments is zero in this frame of reference, the second being that the total final KE of the fragments must add to 900 J. You will have the velocities of each fragment in this frame; add back the 100 m/sec velocity of the center-of-mass and you'll have the velocities of the fragments in the "ground" or stationary reference frame.
 
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  • #3
I think i get what you're saying, and it makes sense but I am for some reason still arriving at the wrong answer. can you check my work.

The Consevation of linear momentum is m1v1=m2v2
but if the total has to equal zero then 0=3x+6y where x and y are their respective velocities. making x=-2y

then for the kinetic energy KEi=KEf and KEf=900
so 0=(0.5*3*x^2)+(0.5*6*y^2)=900
simplifying
x^2+2y^2=600 subsituting the x giving (-2y)^2+ 2y^2=600
giving a y=17.32i? i know i messed up but I am not sure exactly where
 
  • #4
psingh said:
x^2+2y^2=600 subsituting the x giving (-2y)^2+ 2y^2=600

So 4y^2 + 2y^2 = 6y^2 = 600 ,

thus y = 10 m/sec.
 
  • #5
Oh okay that makes sense! Stupid simple mathematical error on my part for some reason i wrote (-2)^2 = -4.

Thanks for all your help! :)
 
  • #6
Good! If you check the speeds in the stationary frame, you'll find that the same 900 J change in kinetic energy has occurred there as well. (Measured values of velocity, momentum, and kinetic energy differ between the frames of reference, but the magnitude of changes in these quantities do not. That's also why magnitudes of accelerations and forces also agree between the reference frames in Galilean relativity.)
The center-of-mass frame, though, can often be the easier one in which to make calculations.
 

1. What are Galilean Transforms and how do they relate to relativity?

Galilean Transforms are mathematical equations that describe the relationship between space and time in classical mechanics. They were developed by Galileo Galilei in the 17th century and were used to explain motion and mechanics before the theory of relativity was introduced by Albert Einstein. Galilean Transforms do not take into account the speed of light, which is a fundamental aspect of relativity.

2. How do Galilean Transforms differ from Lorentz Transforms?

Galilean Transforms and Lorentz Transforms are both mathematical equations used to describe the relationship between space and time. However, Galilean Transforms do not take into account the effects of relativity, such as time dilation and length contraction, while Lorentz Transforms do. Lorentz Transforms are a more accurate description of the relationship between space and time, especially at high speeds.

3. Can Galilean Transforms be used to describe the movement of objects at the speed of light?

No, Galilean Transforms cannot be used to describe the movement of objects at the speed of light. According to the theory of relativity, the speed of light is the maximum speed at which anything can travel. Galilean Transforms do not take this into account and therefore cannot accurately describe the movement of objects at the speed of light.

4. How does the theory of relativity challenge the principles of Galilean Transforms?

The theory of relativity challenges the principles of Galilean Transforms by introducing the concept of the speed of light being constant in all reference frames. This means that the laws of physics, including the relationship between space and time, are the same for all observers, regardless of their relative motion. This contradicts the principles of Galilean Transforms, which assume that the laws of physics are the same for all observers as long as they are moving at a constant speed.

5. Are Galilean Transforms still used in modern science?

While Galilean Transforms are not used to describe the movement of objects at high speeds, they are still used in certain applications where the effects of relativity are negligible. For example, they are still used in classical mechanics for calculating the movement of objects on Earth, where the speed of light is not a factor. However, for more accurate calculations, Lorentz Transforms are typically used instead.

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