![]() | This ![]() It is of interest to the following WikiProjects: | ||||||||||
|
what itrajectory in a simpler explanation
"To neglect the action of the atmosphere, in shaping a trajectory, would (at best) have been considered a futile hypothesis by practical minded investigators, all through the Middle Ages in Europe."
What's this about? Practical-tminded investigation was not the hallmark of the academics of the Middle Ages and early Renaissance. I would have been nice if they'd written mathematical treatises on, for instance, the proper construction of cathedrals (and it would in principle have been possible from the 13th century on), but they didn't; they left that grubby stuff to the trial and error of engineers. (Three guesses who did write the first treatise on mathematical principles of structures. Hint: he's mentioned in the article.) It's true that they wouldn't consider what happens in a vacuum -- because they knew that Nature abnors a vacuum (Aristotle says so); also, the idea of a vacuum is arguably contrary to religion. We could put in that information to mock the poor old Middle Ages, but I think it's more respectful just to zap the passage entirely. -- Dandrake 08:04, 12 December 2005 (UTC)
I received an email flagging a possible problem in the derivation section of "Uphill/downhill in uniform gravity in a vacuum":
My response:
If anyone can help sort this out, that would be greatly appreciated! Samw 04:52, 25 December 2005 (UTC)
Hi Sam,
I have been looking at some references and I think I may know what is going on here. The original reference is known to have many typos (see the book reviews at [2]).
I believe the equation in error should have a cotangent instead of tangent in one of its terms.
I have simulated this situation using an ODE solver and have verified that this equation works and the one in the article does not.
I think I have been able to derive the "Rifleman's Rule" from this expression. It takes a bit of work.
Assume that a rifleman has sighted in his weapon on a flat surface at some range we can call . Because the bullet travels along a parabola, the bore of the weapon will have an angle with respect to the line of sight (LOS). Let's call this angle . We can compute this angle using the following equation.
When the rifleman attempts to shoot uphill with his weapon zeroed at , the gun will actually shoot further than expected (as we will see). Assume that when shooting uphill (hill angle =), the riflebore is at an angle of with respect to gravity. We can write an expression for the bullet point of impact up the hill as follows.
We can approximate using a Taylor approximation as shown below.
After some simplification, we obtain the following expression.
Because the , we see that the bullet will hit the hill a bit further up than where the weapon was zeroed. This range extension can be eliminated by firing the weapon as if it were being aimed at shorter horizontal distance . The following equation illustrates this point.
Does this seem reasonable? Note that the "Rifleman's rule" is an approximation but probably holds well for typical situations.
Mark
Hi Sam,
I agree with you that a step-by-step development is required. I am new to the Wikipedia, so I am still learning how to do things, like adding a figure (which this derivation desperately needs). Also, I really should look at the original reference.
Thanks for the errata reference. With a little more time, I will try to put something more complete together.
For now, I have been able to solve my immediate problem with the ODE solver. However, an equation-based solution would be more useful to the general public. blacksheep 00:57, 27 December 2005 (UTC)
Hi Sam,
I have found the key issue. The definition of in the referenced web article ( [4]) is different than the definition of in your work. With the change of variables and a massive amounts of trigonometric simplification, I have been able to show that your work (minus the one error) and the reference work are equivalent.
I prefer your derivation to the reference article (it uses a coordinate rotation that complicates the algebra). However, the derivation in the reference article does allow a simpler path to the "Rifleman's Rule." blacksheep
Hum, looking through this page there seems to be entirely to much mathematics. Do we really need derivations of formula in an encyclopedia? It might be better just to show key results.
See
Wikipedia:Manual of Style (mathematics)#Proofs and some of this might be bordering on Original research. Its also giving a bit too much away to students who have this set as home work! --
Pfafrich 22:37, 2 January 2006 (UTC)
Its still way to complicated. Heres a simpiler derivation
We know solution is a parabola and hence if we write for the position and for the time we have
the first and second deriv is
At we have
hence
Giving eqn of parabola as
To find the (horizontal) range need to find value of t where z-componant of p is zero i.e.
hence soln are at and subs back in eqn for x component of p gives
By symetry max height will occur when giving
To find derivation in terms of angle of inclination and speed of projectile we have
substituting into eqn for range this gives
Note that calling this polar coordinates is not strictly correct, we still use cartesian coords but just express initial conditions in terms of speed and angle.
Really thats all which is needed for entire derivation, the rest can easily be dropped.
To find the rifle mans range we wish to find the intersection of with the curve
where is the angle of the hill. (I've parametrised the line by t to make things simpler). Equating the z-components and dividing through by t (OK as t=0 is starting soln)
hence
this gives
the length of this is
using the trig identity gives
Divide by gives
Finally
so
So I agree its cot not tan. -- Pfafrich 14:06, 3 January 2006 (UTC)
Its a bit late at night for me, so apoligies if this is giberish.
Still trying to workout what riflemans rule is all about,
(I know some maths but nothing about shooting). One point did occur is that the
a good aproximation is when angle is given in radians.
This is a bit more acurate than just saying its zero. This may help some.
I'll have a bash at tiding up the main article tomorrow. Yep it does seem like wikibook would be a good home. -- Pfafrich 00:22, 4 January 2006 (UTC)
Hi folks,
When Sam and I were discussing this page, I did put together a derivation of the rifleman's rule based on this cotangent version. I have not updated the article because I was concerned that it was a bit much. I have put together a web page that summarizes my work( [5]). I am sure it could be simplified, but it reflects how my head was working.
blacksheep 22:49, 5 January 2006 (UTC)
I like it, I certainly some pictures would go well in the main article. A little quible with using d normally stands for infinitesimal derivative. Here you really using it as a difference might be more appropriate, as is often used for differences.
I've not looked through the maths yet. I'll give it a ponder.
Maybe it time for a new page Riflemans rule, it seems excessive for trajectories. Possible wikibooks? -- Pfafrich 00:49, 6 January 2006 (UTC)
Hi folks,
I have never written a Wikibook, but would like to learn. The "rifleman's rule" does seem to be a well known rule that is missing a good, generally available derivation.
blacksheep 04:35, 6 January 2006 (UTC)
Theres a reciently created page Rifleman's rule wondering if we should trim this page a bit, to save repeated material. -- Salix alba ( talk) 14:32, 31 March 2006 (UTC) (was PfafRich)
Hello folks I would like to add this link to the article: Projectile Lab. It's a JavaScript based trajectory simulator I wrote that asks the user to calculate aspects of the projectile. Any objections? — Edward Z. Yang( Talk) 21:30, 28 December 2006 (UTC)
Yow. This page is nearly incomprehensible. Consider doing a rewrite using more English and less math.
The math itself is not consistent. Equation 11 shows cotangent of theta; all the following equations use tangent theta. I assume that cotangent theta is correct, since in the limiting case alpha=theta, the range should be zero. Thus, it looks like everything after equation 11 is wrong.
(looking up at the talk before my post, it looks like you already understand this, but did not correct the page)
I think that a difficulty is that you have never defined whether theta is measured relative to the horizontal, or relative to the surface. I will presume that it is defined relative to the horizontal.
I'm going to delete all the sections that have tan theta instead of cot theta, which is the part between equation 11 and the rifleman's rule. I'm leaving in the rifleman's rule, since I assume it's correct, even though it's not clear how it comes from equation 11.
I find this sentence very difficult to follow: Thus if the shooter attempts to hit the level distance R, s/he will actually hit the slant target. "In other words, pretend that the inclined target is at a horizontal distance equal to the slant range distance multiplied by the cosine of the inclination angle, and aim as if the target were really at that horizontal position."
What in the world are you saying here? The article is about how far a projectile travels. Geoffrey.landis 03:32, 19 March 2007 (UTC)
Can we consider adding the following to the "Angle of elevation" section? I don't know how exactly to fit it in, but it is an equation solving for angle with differing initial and final heights. g is acceleration of gravity (e.g. 9.8m/s), v is initial velocity, R is desired range, and Delta h is final height minus initial height. It is self-derived and tested, although I am sure someone more important has derived this before. Unfortunately, it would be almost impossible to find an equation like this on google. Geogriffin 23:33, 29 April 2007 (UTC)
...by anticipating the existence of the vacuum...
Calculating as if the object moves through a vacuum is not the same as anticipating the existance thereof. Rather, what is done is ignoring the friction from the air (drag), since the contribution thereof is often small and the formulae are a lot easier without it. Shinobu 11:04, 12 May 2007 (UTC)
Thus Rs / R is a positive value meaning the range downhill is always further than along level terrain. This makes perfect sense as it is expected that gravity will assist the projectile, giving it greater range.
Isn't the range assisted by the fact that the ground is simply further away? That is, the lower ground gives the projectile more space to drop, which extends the time it is in the air, allowing it to travel further in the x direction. As far as I know gravity doesn't affect horizontal travel aside from limiting air time.
MagLab ( talk) 00:21, 1 August 2008 (UTC)
I would like to have some information about real rocket trajectories. It would be intersteing to see which flight path real rocket systems use. greets, Andreas —Preceding unsigned comment added by 84.128.60.211 ( talk) 20:57, 25 October 2008 (UTC)
I understand presenting the differential equations that model projectile motion in uniform gravity - but wonder of what help such a contrived derivation could of the equation for the trajectory would be of to the reader. Again the derivation presented assumes a lot of background knowedge - such as "path of the projectile is known to be a parabola"! Well perhaps the editor meant the path is "known" to be a parabola.. given that differntial equations that model it is such.. Anyway.. I propose replacing it with a more direct/simple/elegant algebraic or perhaps even a visual one.
Dilip rajeev ( talk) 17:49, 9 March 2009 (UTC)
I am replacing this unnecessarily contrived and incomplete derivation with a simpler one - the reasoning of which I outline below:
Assume the point of launch as the Origin of the right-hand co-ordinate axes - with the x axis along the ground and y along vertical. Seen from a free-fall frame, which at t=0 is at (x,y)=(0,0) - the object would take a path given by y=xtan(thta). The co-ordnates of this free fall frame is given by y=-1/2gt^2 .. where t=x/(v*Cos(thta)). So translating the co-ordinates back to our original frame, we have ....
Dilip rajeev ( talk) 17:49, 9 March 2009 (UTC)
This article, as well as the projectile motion article, needs a lot of work, IMHO. I have written a good bit of material for high-school and first-year college students (AP Physics) on the subject of projectile motion. Perhaps I can find the time to adapt some of those handouts to make a more accessible and better-organized article...
Math derivations are loads of fun, I've enjoyed doing that sort of thing for some 40 years, but not in an encyclopedia article (nor in most published articles). If the derivation is important, summarize it in the text and put the details in an appendix. Keep the ideas flowing, don't bog down with math, especially when writing for a general, non-specialist audience. Last I heard, an encyclopedia is not a math, nor physics, textbook... (although, full disclosure, in "my" article on Experimental uncertainty analysis I did put in a derivation of the POE expression, just because it is central to the article and is difficult to find).
For a far more detailed (definitely) and accessible (maybe) treatment of this material, you might like to go to this site and scroll down to and click on the /nikenuke directory. Once in there, click the /projectilePDF directory and a whole bunch of PDF papers will be available. These papers will provide a lot of analysis, some of which is beyond high-school level, but several of them should still be useful. These are not the handouts I mentioned above, those are separate; these are intended more for teachers, to provide the mathematical background details as needed for class preparation. Also, there's a couple of Java sims on projectiles in the /nikenuke directory- click on the respective HTML files. Rb88guy ( talk) 01:59, 25 September 2009 (UTC)
The article stated: "Physically, this corresponds to a direct shot versus a mortar shot up and over obstacles to the target."
This notion is incorrect. There are two initial angles, theta_0_a and theta_0_b, that have the same range. It makes no difference whether a mortar or a rifle fires the shot. The notion of direct versus indirect fires has no place in kinematics--both are examples of projectile motion, and both are governed by the same rules. "Indirect" and "direct" are military terms, not physics terms.
In reality, maximum range for a given initial velocity occurs when the initial angle of elevation is 45 degrees. In that case, there is only one angle that gives the same range, i.e. 45 degrees. At all other angles of elevation from 0-90 degrees, there are two angles which will give the same range. For those two angles, the absolute value of the difference between said angle and 45 degrees is the same. For example, if all else is equal, and one initial angle is 53 degrees, the absolute value of the difference is 8 degrees. Therefore the corresponding angle is 45-8=37 degrees. So both initial angles 53 degrees and 37 degrees would result in the same range.
Back to the mistaken notion: If you fire your rifle at a 10 degree angle of fire, you could achieve the same range by firing the rifle at an 80 degree angle, but have a plunging trajectory rather than a more direct trajectory, though they would both be parabolas. Of course, this all neglects air resistance, but that isn't covered in this section. Further, when you shoot a rifle, your target is not at the maximum range, which is what the angle of elevation section deals with.
Marktaff ( talk) 08:28, 20 October 2010 (UTC)
The range section ties the range to sin(2theta) is this right? Why 2? Sorry for lack of symbols.
As an impartial observer who just happened upon this page, I wonder if we still need to have a warning that the factual accuracy is disputed for Equation 11. The derivation provided in the Uphill/downhill in uniform gravity in a vacuum section is based on well-known physics principles, and the math is easy enough for an advanced high school student to comprehend. The warning is not meant to last forever, and it has been almost 10 years since the warning was put in place. I vote to remove it. Jdlawlis ( talk) 01:34, 15 October 2017 (UTC)
motion of the arrow of dartboard M shoaib Akhtar ( talk) 16:56, 19 October 2019 (UTC)
![]() | This ![]() It is of interest to the following WikiProjects: | ||||||||||
|
what itrajectory in a simpler explanation
"To neglect the action of the atmosphere, in shaping a trajectory, would (at best) have been considered a futile hypothesis by practical minded investigators, all through the Middle Ages in Europe."
What's this about? Practical-tminded investigation was not the hallmark of the academics of the Middle Ages and early Renaissance. I would have been nice if they'd written mathematical treatises on, for instance, the proper construction of cathedrals (and it would in principle have been possible from the 13th century on), but they didn't; they left that grubby stuff to the trial and error of engineers. (Three guesses who did write the first treatise on mathematical principles of structures. Hint: he's mentioned in the article.) It's true that they wouldn't consider what happens in a vacuum -- because they knew that Nature abnors a vacuum (Aristotle says so); also, the idea of a vacuum is arguably contrary to religion. We could put in that information to mock the poor old Middle Ages, but I think it's more respectful just to zap the passage entirely. -- Dandrake 08:04, 12 December 2005 (UTC)
I received an email flagging a possible problem in the derivation section of "Uphill/downhill in uniform gravity in a vacuum":
My response:
If anyone can help sort this out, that would be greatly appreciated! Samw 04:52, 25 December 2005 (UTC)
Hi Sam,
I have been looking at some references and I think I may know what is going on here. The original reference is known to have many typos (see the book reviews at [2]).
I believe the equation in error should have a cotangent instead of tangent in one of its terms.
I have simulated this situation using an ODE solver and have verified that this equation works and the one in the article does not.
I think I have been able to derive the "Rifleman's Rule" from this expression. It takes a bit of work.
Assume that a rifleman has sighted in his weapon on a flat surface at some range we can call . Because the bullet travels along a parabola, the bore of the weapon will have an angle with respect to the line of sight (LOS). Let's call this angle . We can compute this angle using the following equation.
When the rifleman attempts to shoot uphill with his weapon zeroed at , the gun will actually shoot further than expected (as we will see). Assume that when shooting uphill (hill angle =), the riflebore is at an angle of with respect to gravity. We can write an expression for the bullet point of impact up the hill as follows.
We can approximate using a Taylor approximation as shown below.
After some simplification, we obtain the following expression.
Because the , we see that the bullet will hit the hill a bit further up than where the weapon was zeroed. This range extension can be eliminated by firing the weapon as if it were being aimed at shorter horizontal distance . The following equation illustrates this point.
Does this seem reasonable? Note that the "Rifleman's rule" is an approximation but probably holds well for typical situations.
Mark
Hi Sam,
I agree with you that a step-by-step development is required. I am new to the Wikipedia, so I am still learning how to do things, like adding a figure (which this derivation desperately needs). Also, I really should look at the original reference.
Thanks for the errata reference. With a little more time, I will try to put something more complete together.
For now, I have been able to solve my immediate problem with the ODE solver. However, an equation-based solution would be more useful to the general public. blacksheep 00:57, 27 December 2005 (UTC)
Hi Sam,
I have found the key issue. The definition of in the referenced web article ( [4]) is different than the definition of in your work. With the change of variables and a massive amounts of trigonometric simplification, I have been able to show that your work (minus the one error) and the reference work are equivalent.
I prefer your derivation to the reference article (it uses a coordinate rotation that complicates the algebra). However, the derivation in the reference article does allow a simpler path to the "Rifleman's Rule." blacksheep
Hum, looking through this page there seems to be entirely to much mathematics. Do we really need derivations of formula in an encyclopedia? It might be better just to show key results.
See
Wikipedia:Manual of Style (mathematics)#Proofs and some of this might be bordering on Original research. Its also giving a bit too much away to students who have this set as home work! --
Pfafrich 22:37, 2 January 2006 (UTC)
Its still way to complicated. Heres a simpiler derivation
We know solution is a parabola and hence if we write for the position and for the time we have
the first and second deriv is
At we have
hence
Giving eqn of parabola as
To find the (horizontal) range need to find value of t where z-componant of p is zero i.e.
hence soln are at and subs back in eqn for x component of p gives
By symetry max height will occur when giving
To find derivation in terms of angle of inclination and speed of projectile we have
substituting into eqn for range this gives
Note that calling this polar coordinates is not strictly correct, we still use cartesian coords but just express initial conditions in terms of speed and angle.
Really thats all which is needed for entire derivation, the rest can easily be dropped.
To find the rifle mans range we wish to find the intersection of with the curve
where is the angle of the hill. (I've parametrised the line by t to make things simpler). Equating the z-components and dividing through by t (OK as t=0 is starting soln)
hence
this gives
the length of this is
using the trig identity gives
Divide by gives
Finally
so
So I agree its cot not tan. -- Pfafrich 14:06, 3 January 2006 (UTC)
Its a bit late at night for me, so apoligies if this is giberish.
Still trying to workout what riflemans rule is all about,
(I know some maths but nothing about shooting). One point did occur is that the
a good aproximation is when angle is given in radians.
This is a bit more acurate than just saying its zero. This may help some.
I'll have a bash at tiding up the main article tomorrow. Yep it does seem like wikibook would be a good home. -- Pfafrich 00:22, 4 January 2006 (UTC)
Hi folks,
When Sam and I were discussing this page, I did put together a derivation of the rifleman's rule based on this cotangent version. I have not updated the article because I was concerned that it was a bit much. I have put together a web page that summarizes my work( [5]). I am sure it could be simplified, but it reflects how my head was working.
blacksheep 22:49, 5 January 2006 (UTC)
I like it, I certainly some pictures would go well in the main article. A little quible with using d normally stands for infinitesimal derivative. Here you really using it as a difference might be more appropriate, as is often used for differences.
I've not looked through the maths yet. I'll give it a ponder.
Maybe it time for a new page Riflemans rule, it seems excessive for trajectories. Possible wikibooks? -- Pfafrich 00:49, 6 January 2006 (UTC)
Hi folks,
I have never written a Wikibook, but would like to learn. The "rifleman's rule" does seem to be a well known rule that is missing a good, generally available derivation.
blacksheep 04:35, 6 January 2006 (UTC)
Theres a reciently created page Rifleman's rule wondering if we should trim this page a bit, to save repeated material. -- Salix alba ( talk) 14:32, 31 March 2006 (UTC) (was PfafRich)
Hello folks I would like to add this link to the article: Projectile Lab. It's a JavaScript based trajectory simulator I wrote that asks the user to calculate aspects of the projectile. Any objections? — Edward Z. Yang( Talk) 21:30, 28 December 2006 (UTC)
Yow. This page is nearly incomprehensible. Consider doing a rewrite using more English and less math.
The math itself is not consistent. Equation 11 shows cotangent of theta; all the following equations use tangent theta. I assume that cotangent theta is correct, since in the limiting case alpha=theta, the range should be zero. Thus, it looks like everything after equation 11 is wrong.
(looking up at the talk before my post, it looks like you already understand this, but did not correct the page)
I think that a difficulty is that you have never defined whether theta is measured relative to the horizontal, or relative to the surface. I will presume that it is defined relative to the horizontal.
I'm going to delete all the sections that have tan theta instead of cot theta, which is the part between equation 11 and the rifleman's rule. I'm leaving in the rifleman's rule, since I assume it's correct, even though it's not clear how it comes from equation 11.
I find this sentence very difficult to follow: Thus if the shooter attempts to hit the level distance R, s/he will actually hit the slant target. "In other words, pretend that the inclined target is at a horizontal distance equal to the slant range distance multiplied by the cosine of the inclination angle, and aim as if the target were really at that horizontal position."
What in the world are you saying here? The article is about how far a projectile travels. Geoffrey.landis 03:32, 19 March 2007 (UTC)
Can we consider adding the following to the "Angle of elevation" section? I don't know how exactly to fit it in, but it is an equation solving for angle with differing initial and final heights. g is acceleration of gravity (e.g. 9.8m/s), v is initial velocity, R is desired range, and Delta h is final height minus initial height. It is self-derived and tested, although I am sure someone more important has derived this before. Unfortunately, it would be almost impossible to find an equation like this on google. Geogriffin 23:33, 29 April 2007 (UTC)
...by anticipating the existence of the vacuum...
Calculating as if the object moves through a vacuum is not the same as anticipating the existance thereof. Rather, what is done is ignoring the friction from the air (drag), since the contribution thereof is often small and the formulae are a lot easier without it. Shinobu 11:04, 12 May 2007 (UTC)
Thus Rs / R is a positive value meaning the range downhill is always further than along level terrain. This makes perfect sense as it is expected that gravity will assist the projectile, giving it greater range.
Isn't the range assisted by the fact that the ground is simply further away? That is, the lower ground gives the projectile more space to drop, which extends the time it is in the air, allowing it to travel further in the x direction. As far as I know gravity doesn't affect horizontal travel aside from limiting air time.
MagLab ( talk) 00:21, 1 August 2008 (UTC)
I would like to have some information about real rocket trajectories. It would be intersteing to see which flight path real rocket systems use. greets, Andreas —Preceding unsigned comment added by 84.128.60.211 ( talk) 20:57, 25 October 2008 (UTC)
I understand presenting the differential equations that model projectile motion in uniform gravity - but wonder of what help such a contrived derivation could of the equation for the trajectory would be of to the reader. Again the derivation presented assumes a lot of background knowedge - such as "path of the projectile is known to be a parabola"! Well perhaps the editor meant the path is "known" to be a parabola.. given that differntial equations that model it is such.. Anyway.. I propose replacing it with a more direct/simple/elegant algebraic or perhaps even a visual one.
Dilip rajeev ( talk) 17:49, 9 March 2009 (UTC)
I am replacing this unnecessarily contrived and incomplete derivation with a simpler one - the reasoning of which I outline below:
Assume the point of launch as the Origin of the right-hand co-ordinate axes - with the x axis along the ground and y along vertical. Seen from a free-fall frame, which at t=0 is at (x,y)=(0,0) - the object would take a path given by y=xtan(thta). The co-ordnates of this free fall frame is given by y=-1/2gt^2 .. where t=x/(v*Cos(thta)). So translating the co-ordinates back to our original frame, we have ....
Dilip rajeev ( talk) 17:49, 9 March 2009 (UTC)
This article, as well as the projectile motion article, needs a lot of work, IMHO. I have written a good bit of material for high-school and first-year college students (AP Physics) on the subject of projectile motion. Perhaps I can find the time to adapt some of those handouts to make a more accessible and better-organized article...
Math derivations are loads of fun, I've enjoyed doing that sort of thing for some 40 years, but not in an encyclopedia article (nor in most published articles). If the derivation is important, summarize it in the text and put the details in an appendix. Keep the ideas flowing, don't bog down with math, especially when writing for a general, non-specialist audience. Last I heard, an encyclopedia is not a math, nor physics, textbook... (although, full disclosure, in "my" article on Experimental uncertainty analysis I did put in a derivation of the POE expression, just because it is central to the article and is difficult to find).
For a far more detailed (definitely) and accessible (maybe) treatment of this material, you might like to go to this site and scroll down to and click on the /nikenuke directory. Once in there, click the /projectilePDF directory and a whole bunch of PDF papers will be available. These papers will provide a lot of analysis, some of which is beyond high-school level, but several of them should still be useful. These are not the handouts I mentioned above, those are separate; these are intended more for teachers, to provide the mathematical background details as needed for class preparation. Also, there's a couple of Java sims on projectiles in the /nikenuke directory- click on the respective HTML files. Rb88guy ( talk) 01:59, 25 September 2009 (UTC)
The article stated: "Physically, this corresponds to a direct shot versus a mortar shot up and over obstacles to the target."
This notion is incorrect. There are two initial angles, theta_0_a and theta_0_b, that have the same range. It makes no difference whether a mortar or a rifle fires the shot. The notion of direct versus indirect fires has no place in kinematics--both are examples of projectile motion, and both are governed by the same rules. "Indirect" and "direct" are military terms, not physics terms.
In reality, maximum range for a given initial velocity occurs when the initial angle of elevation is 45 degrees. In that case, there is only one angle that gives the same range, i.e. 45 degrees. At all other angles of elevation from 0-90 degrees, there are two angles which will give the same range. For those two angles, the absolute value of the difference between said angle and 45 degrees is the same. For example, if all else is equal, and one initial angle is 53 degrees, the absolute value of the difference is 8 degrees. Therefore the corresponding angle is 45-8=37 degrees. So both initial angles 53 degrees and 37 degrees would result in the same range.
Back to the mistaken notion: If you fire your rifle at a 10 degree angle of fire, you could achieve the same range by firing the rifle at an 80 degree angle, but have a plunging trajectory rather than a more direct trajectory, though they would both be parabolas. Of course, this all neglects air resistance, but that isn't covered in this section. Further, when you shoot a rifle, your target is not at the maximum range, which is what the angle of elevation section deals with.
Marktaff ( talk) 08:28, 20 October 2010 (UTC)
The range section ties the range to sin(2theta) is this right? Why 2? Sorry for lack of symbols.
As an impartial observer who just happened upon this page, I wonder if we still need to have a warning that the factual accuracy is disputed for Equation 11. The derivation provided in the Uphill/downhill in uniform gravity in a vacuum section is based on well-known physics principles, and the math is easy enough for an advanced high school student to comprehend. The warning is not meant to last forever, and it has been almost 10 years since the warning was put in place. I vote to remove it. Jdlawlis ( talk) 01:34, 15 October 2017 (UTC)
motion of the arrow of dartboard M shoaib Akhtar ( talk) 16:56, 19 October 2019 (UTC)