Classical vs. relativistic
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That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
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Exercise:
Use the classical expression for kinetic energy to calculate the speed of an electron with a kinetic energy of abcliste abc EaO and abc EbO. abcliste Compare with the results found with the relativistic expression.
Solution:
textbfClassical expression: From E_k frac m_e v^ it follows that v_mathrmcl sqrtfracE_km_e csqrtfracE_km_e c^ csqrtfracE_kE_ with the rest energy of the electron E_ m_e c^ approx ErP. textbfRelativistic expression: The kinetic energy is the difference between total energy and rest energy E_k E-E_ gamma-E_ Hence gamma +fracE_kE_ and with gamma /sqrt-v^/c^ the speed is v_mathrmrel csqrt-fracgamma^ abcliste abc For E_aEaO we find with the classical expression v_mathrmcl csqrtfracE_aE_ ncctimessqrtfractimesEaOErP vca approx resultvcaP which corresponds to v_mathrmcl/c bca. The relativistic calculation yields gamma +fracE_aE_ +fracEaOErP ga v_mathrmrel csqrt-fracgamma^ ncctimessqrt-fracleftgaright^ vea approx resultveaP i.e. v_mathrmrel/c bea. The classical speed is therefore too large by fracv_mathrmclv_mathrmrel- dva approx resultdvaP At this energy the classical expression is still a good approximation. abc For E_bEbO we find with the classical expression v_mathrmcl csqrtfracE_bE_ ncctimessqrtfractimesEbOErP vcb approx resultvcbP which corresponds to v_mathrmcl/c bcb i.e. a speed much larger than the speed of light. The classical result is clearly wrong. The relativistic calculation yields gamma +fracE_bE_ +fracEbOErP gb v_mathrmrel csqrt-fracgamma^ ncctimessqrt-fracleftgbright^ veb approx resultvebP i.e. v_mathrmrel/c beb: the electron approaches the speed of light but never reaches it. The classical speed is too large by fracv_mathrmclv_mathrmrel- dvb approx resultdvbP abcliste
Use the classical expression for kinetic energy to calculate the speed of an electron with a kinetic energy of abcliste abc EaO and abc EbO. abcliste Compare with the results found with the relativistic expression.
Solution:
textbfClassical expression: From E_k frac m_e v^ it follows that v_mathrmcl sqrtfracE_km_e csqrtfracE_km_e c^ csqrtfracE_kE_ with the rest energy of the electron E_ m_e c^ approx ErP. textbfRelativistic expression: The kinetic energy is the difference between total energy and rest energy E_k E-E_ gamma-E_ Hence gamma +fracE_kE_ and with gamma /sqrt-v^/c^ the speed is v_mathrmrel csqrt-fracgamma^ abcliste abc For E_aEaO we find with the classical expression v_mathrmcl csqrtfracE_aE_ ncctimessqrtfractimesEaOErP vca approx resultvcaP which corresponds to v_mathrmcl/c bca. The relativistic calculation yields gamma +fracE_aE_ +fracEaOErP ga v_mathrmrel csqrt-fracgamma^ ncctimessqrt-fracleftgaright^ vea approx resultveaP i.e. v_mathrmrel/c bea. The classical speed is therefore too large by fracv_mathrmclv_mathrmrel- dva approx resultdvaP At this energy the classical expression is still a good approximation. abc For E_bEbO we find with the classical expression v_mathrmcl csqrtfracE_bE_ ncctimessqrtfractimesEbOErP vcb approx resultvcbP which corresponds to v_mathrmcl/c bcb i.e. a speed much larger than the speed of light. The classical result is clearly wrong. The relativistic calculation yields gamma +fracE_bE_ +fracEbOErP gb v_mathrmrel csqrt-fracgamma^ ncctimessqrt-fracleftgbright^ veb approx resultvebP i.e. v_mathrmrel/c beb: the electron approaches the speed of light but never reaches it. The classical speed is too large by fracv_mathrmclv_mathrmrel- dvb approx resultdvbP abcliste
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Exercise:
Use the classical expression for kinetic energy to calculate the speed of an electron with a kinetic energy of abcliste abc EaO and abc EbO. abcliste Compare with the results found with the relativistic expression.
Solution:
textbfClassical expression: From E_k frac m_e v^ it follows that v_mathrmcl sqrtfracE_km_e csqrtfracE_km_e c^ csqrtfracE_kE_ with the rest energy of the electron E_ m_e c^ approx ErP. textbfRelativistic expression: The kinetic energy is the difference between total energy and rest energy E_k E-E_ gamma-E_ Hence gamma +fracE_kE_ and with gamma /sqrt-v^/c^ the speed is v_mathrmrel csqrt-fracgamma^ abcliste abc For E_aEaO we find with the classical expression v_mathrmcl csqrtfracE_aE_ ncctimessqrtfractimesEaOErP vca approx resultvcaP which corresponds to v_mathrmcl/c bca. The relativistic calculation yields gamma +fracE_aE_ +fracEaOErP ga v_mathrmrel csqrt-fracgamma^ ncctimessqrt-fracleftgaright^ vea approx resultveaP i.e. v_mathrmrel/c bea. The classical speed is therefore too large by fracv_mathrmclv_mathrmrel- dva approx resultdvaP At this energy the classical expression is still a good approximation. abc For E_bEbO we find with the classical expression v_mathrmcl csqrtfracE_bE_ ncctimessqrtfractimesEbOErP vcb approx resultvcbP which corresponds to v_mathrmcl/c bcb i.e. a speed much larger than the speed of light. The classical result is clearly wrong. The relativistic calculation yields gamma +fracE_bE_ +fracEbOErP gb v_mathrmrel csqrt-fracgamma^ ncctimessqrt-fracleftgbright^ veb approx resultvebP i.e. v_mathrmrel/c beb: the electron approaches the speed of light but never reaches it. The classical speed is too large by fracv_mathrmclv_mathrmrel- dvb approx resultdvbP abcliste
Use the classical expression for kinetic energy to calculate the speed of an electron with a kinetic energy of abcliste abc EaO and abc EbO. abcliste Compare with the results found with the relativistic expression.
Solution:
textbfClassical expression: From E_k frac m_e v^ it follows that v_mathrmcl sqrtfracE_km_e csqrtfracE_km_e c^ csqrtfracE_kE_ with the rest energy of the electron E_ m_e c^ approx ErP. textbfRelativistic expression: The kinetic energy is the difference between total energy and rest energy E_k E-E_ gamma-E_ Hence gamma +fracE_kE_ and with gamma /sqrt-v^/c^ the speed is v_mathrmrel csqrt-fracgamma^ abcliste abc For E_aEaO we find with the classical expression v_mathrmcl csqrtfracE_aE_ ncctimessqrtfractimesEaOErP vca approx resultvcaP which corresponds to v_mathrmcl/c bca. The relativistic calculation yields gamma +fracE_aE_ +fracEaOErP ga v_mathrmrel csqrt-fracgamma^ ncctimessqrt-fracleftgaright^ vea approx resultveaP i.e. v_mathrmrel/c bea. The classical speed is therefore too large by fracv_mathrmclv_mathrmrel- dva approx resultdvaP At this energy the classical expression is still a good approximation. abc For E_bEbO we find with the classical expression v_mathrmcl csqrtfracE_bE_ ncctimessqrtfractimesEbOErP vcb approx resultvcbP which corresponds to v_mathrmcl/c bcb i.e. a speed much larger than the speed of light. The classical result is clearly wrong. The relativistic calculation yields gamma +fracE_bE_ +fracEbOErP gb v_mathrmrel csqrt-fracgamma^ ncctimessqrt-fracleftgbright^ veb approx resultvebP i.e. v_mathrmrel/c beb: the electron approaches the speed of light but never reaches it. The classical speed is too large by fracv_mathrmclv_mathrmrel- dvb approx resultdvbP abcliste
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Lecture notes Relativity by by
| Title | Matched on |
|---|---|
| Relativistic proton | formula |