Burglars and BBs: Newsletter Challenge (04/03/07)

g = 9.81ms^-1

V = 60m/s

V_x = V(100/√10400)

V_y = V(20/√10400)

Δt = Δx/V_x = 1.70s

H_r = 20 - ½.g.t²

H_rt = 5.83m

H_bb = V_y.t - ½.g.t²

H_bbt = 5.44m

H_rt - H_bbt = 0.38m

At a first look I'd say: Yes I can hit him, But if I aimed at his chest then I'd hit him in the head. Of course this depends on a trillion other factors such as: Can we ignore air effects (drag and crosswind)? Can we identify burgulars at 100m away? Can our BB gun shoot that straight? Do 'good citizens' carry BB guns around with them? Does it matter whether we hit him or not, given that he is falling 20m? Is the ground flat between you and the building? Is the question being asked more than 1.7s after I release the BB gun? How far out from the building does the burgular jump? Does all this happen on Earth (someone always has to ask)? What if a pig flies between you and the burgular?

How do you get in so quick . I haven't even read the Q yet!

Trying to be a good citizen you shoot the burglar. Oh yeah !! Great citizen. What´s next... Johnny was walking down the street and sees and old lady with her little poodle. The little poodle goes and takes a pis near the side walk by a tree. Johnny, the good citizen, take up his 12 gauge and blows the lady´s head off...

Can you say paranoia! I'm sure if it was your place you would be thanking the guy. Take it down a notch!!!

What we have here is a person who believes that it is OK to be a burglar but not OK to prevent crime... Folks should consider that these questions are not for people whose IQ does not exceed their shoe size.

Sounds right to me. Whats the problem?

Davo,

If you aim for the burglar's chest, you will hit the chest (not his head). See #5 by Uncle Red for an example on this. This works because the BB and Burglar are both being drawn towards the earth's center at the same rate and the flight of the BB is not straight but curved.

Therefore, if both burglar and BB begin their decent at the same instant, the vertical drop of each will be equal and where you aim is where you hit.

Davo,

.....Sorry, I ment to say See #4 by Uncle Red and not #5 for the example (sorry Uncle Red as well) .....

.... my #5 is wrong ? Explain please . Davo has convinced me further thought may be redundant , did I jot that off I haste? I 've figured Davo generally gets in quick with the best answer , and the rest is superfluous . Hence my quick unexplained formula . Can you correct me or just say "Davo has done it ". And how does he find time to even add a diagram . Respect !

Hi Kris,

Not saying #5 is wrong ...... just accidentally referenced it when responding to #1.

Please read #6 and #7 together and you'll get the flow .....

I appreciate the courtesy Guest. I think the credit here will be Davo's. I'm running on an equation I was too lazy to work out + the knowledge that Davo has probably said all the important points while everyone else was sleeping. I like a good debatable question , but this one seems to have been nailed.

Yes, I see you and Uncle Red are right. Subtracting equation

H_r = 20 - ½.g.t² from H_bb = V_y.t - ½.g.t²

gives ΔH = Δy - V_y.t

= Δy-[V.(Δy/(Δx²+Δy²)^½)].[Δx.(Δx²+Δy²)^½/(V.Δx)]

= Δy-Δy = 0

Also I got a bit carried away and said that it had to be on Earth. Not so. The same reasoning and equation applies for any gravitational acceleration, provided we can ignore curvature.

Hi Davo,

I have a feeling that your original answer was right! If the BB gun was adjusted so that it would hit where aimed at 100m range (if that's possible), then it will indeed hit the falling person at a higher point than aimed at. The vertical travel of the bullet has already been taken out by the range setting.

Or am I missing something, assuming that air resistance and deflection's are ignored?

Regards, Jorrie

Hi Jorrie,

If it was set to hit the target at a particular then range then it would hit that point wouldn't it? So it would miss the burgular by several metres and he'd only suffer the consequences of his 20m fall.

Hi Davo,

Yep, one of those very undefined challenges; amongst many other definitions one needs to know is, what was meant by "point the BB gun"? Aim it properly or bore sight it?

Regards, Jorrie

Hi Jorrie, I can't comment on this one. I generally try to find other ways of expressing my good citizenship, rather than walking the streets as a BB vigilante, Davo

When the burglar saw a gun pointed at him, he jumped off the building on the opposite side as the shooter. It doesn't matter if the barrel was pointed at the burglar or pointed whereas the BB would hit the burglar on the roof, since it would fly over or hit the side of the building, not hitting the burglar either way.

The burglar then went through the canvas awning on the way down, breaking the fall, and escaped safely to burgle another day.

Hi Davo,

You and Jorie (in the post just before this) are completely correct.

In answering many questions, one must make some assumptions. But to be fair to the question writer, the assumptions should at least be reasonable. If the shooter points the gun at the robber and pulls the trigger, then the reasonable assumption is that he was trying to shoot the robber. Clearly, he wasn't just waving the rifle around randomly. If he is aiming right-left (necessary for success of the logic proposed in the official answer) then you can assume he is also aiming up-down.

So the shooter doesn't need to be a physics whiz, just the possessor of the ordinary intellegence of a hunter, or a cop, or a kid with a bb gun, all of whom (like a basket ball player) adjust the trajectory initiation to account for the effects of gravity. (Long shot, aim higher.) Try to tell a cop that if he aims at the robber, his bullet will drop well below the target, and that in order for him to hit the target, the robber must jump off the roof!

To interpret the question in the alternative way means that you are assuming that the shooter has bb gun, but that he never learned to shoot it, and that even if he is new to shooting (unlikely though that may be) that he has also never seen a basketball shot, or a soccer kick, or a football pass.

This is a classic mis-worded and misinterpreted physics text question (usually, a monkey drops from a tree, when he sees the flash from the gun.) The student who has retained a shred of common sense is marked wrong. The student who can simply do the math (although none is required here) but doesn't really grasp the concepts involved, gets full marks.

Sure there are reasonable simplifying assumptions that one employees in physics discussions. These really should be spelled out in the question, as they usually are ("Ignoring the effects of air resistance, ... " etc.) But to start with unreasonable assumptions (such as the possessor of the gun has no idea how to use it) defies common sense. Certainly I'm getting crotchety in my old age, but it seems to me that the application of common sense is becoming rare. In physics, especially, there is a huge difference between simply doing the math, and actually grasping the concept. My fear is that our schools are turning out people who can only do the math.

Physics is not some theoretical construct with application in only the most abstract ways. Physics describes how things really work. Question/answer combos that can only make sense if one ignores how things really work erode science education.

The physics class demo, in which a tube's centerline is aligned with a target (and then the shot and target-drop occur simultaneously) is what this question is attempting to put into real-world context -- but the situations are obviously different.

Rant, rant, rant.

. . . And provided you and the burglar are not traveling at the speed of light were Lorenz says at least the length of axis of travel changes . . .

I love these challenges!

Zeke

Surely that is irrelevant if you and the burglar are travelling the same speed before he starts to fall?

Your not factoring in momentum? the BB though affected by gravity will not fall at 9.8m/s SQRd until it loses all horizontal momentum

Hi dazed and confused,

Vertical and horizontal components of velocity can be treated independently. If we ignore wind resistance then the BB's horizontal velocity will remain unchanged. From the moment the BB is launched the vertical component its will be affected by gravity. Does this make sense?

Davo

Davo,

Of course it make sense!

Abe

Allow me to rephrase: Does that make sense to you, Dazed and Confused? Davo

The sight on the gun is set to a range compensating for the fall in the projectile at that range, i would guess 25 yards or so on something small like a BB gun (up to 300 yard for a high velocity hunting rifle). Therefore, the BB would likely hit the target high of the mark (i really do not feel like doing the calculation for the curve intercepting a line, and therefore can not strongly substantiate this).

Only if the gun barrel is horizontal and at the same height as the guys head when you fire, so the BB will track him down as they both fall at the same rate but as soon as you fire upwards the your BB follows a balistic curve and during almost half of it's flight it is traveling up as the burglar falls down. I don't know the range of a BB gun, possibly not even 100m!? but unless the angle of the gun aiming up gives it a range of exactly that then I think you miss.

Hi Janny, It really doesn't matter whether the gun barrel is horizontal or not. Consider an extreme case where there is no gravity. The burgular would stay still and the projectile would travel in a straight line and hit him. If we add gravity to the situation, then while the burgular is falling towards the earth and accelerating at 9.81ms^-2 the projectile is also deviating from its "zero gravity" straight line trajectory with the same acceleration. Even while the projectile is travelling up it is accelerating downwards at 9.81ms^-2.

The key point is that they both start travelling at the same time so gravity acts on both of them for the same length of time.

I agree that in reality we would probably fall short due to air resistance, but I have a feeling the answer will say "Assuming no losses due to air resistance you would hit him." Davo

The bullet will fall down with the burglar. And at the same velocity.

burglar would hit ground before bb reach him....in real world.

A question is asked at 10.01 , Davo posts at 12.51 . How long......

(Thought I'd set the tone )

I have a time machine. It's called Australia.

This question has a week to run - 1 week at light speed will not be enough to escape. Is the 'answer' to the boat up yet ? Looks like coffee time is back to factorising products of primes.. Post-it notes ; gods gift to the office

Come on Kris, we have to last a few more days: How much will the "coriolis effect" affect the results if you are at 45^o lattitude firing due North.

I may amuse myself by calculating the trajectory to shoot myself in the back by firing around the Globe. Working out the reaction force on myself etc may be difficult. I now wonder if this is an actual possibility by firing just upwards (?).

Silly, me! I thought the time machine was called the International Date Line. You may go forward or backward in time 24 hrs. simply by crossing the line!

Sounds like the classic " Shoot the Dropping Monkey from the Tree" kinda problem.

I recall from years ago a similar experiment in 12th grade physics where the teacher had a blow gun (metal tube) with an electric trip wire on the exit end. He would aim his "blow gun" at a toy monkey that he had suspended from an electro-magnet which was hung on the ceiling. He blew a marble from the tube and the toy monkey dropped at the instant the marble left the tube (tripping the circuit) and he consistantly hit the monkey about a foot from the floor.

It was pretty neat.

hi red........we must be from the same physics class... saw similar setup years ago and it hit the falling object everytime (much to our very young amazement).......Davo is correct and it does not matter were the bb hits, as it must hit where aimed as long as it reachs the person before hitting the ground....as long as all other variables are ignored (air resistance, etc.)...the plain answer to this problem is "YES".

I was waiting for this reply. I too had this experiment in high school physics and as soon as I read the problem the first thing to pop into mind was - "this is the monkey drop test." I brought this problem up a couple of years ago at work and no one had ever heard of it. Glad to see that I'm not nuts!

"...12th grade physics where the teacher had a blow gun (metal tube) with an electric trip wire..."

My dad was the physics teacher at our school. This trick was one of his favorites. My senior year he finally achieved a major goal - he had a relevant demonstration for every single day of the class. Kids started calling him "Trick-a-Day Trinklein". He loved it.

well, today is "take your child to work day" and the engineering department here is ALWAYS the favorite place to visit........maybe because we are just big kids :)

anyway, this sounded like a fun experiment...........SO.........after a couple days of tinkering we made this setup...........and it REALLY works.....

this was not as easy as it looks............

we can vary the air pressure between 40-100 psi and nail the can every time.

http://s21.photobucket.com/albums/b299/mudbilly/gravity/?action=view&current=DSC01562.jpg&slideshow=true&interval=3

video to come later

any questions?

((20- (g.t²/2))² + (100)² = (60 . t)² ? (I'm feeling lazy)

This is a high school physic problem, or first semester engineering dynamics. You are using pythogreans theorem, which is totally wrong in this case. You have to figure the problem as a trajectory/time of travel for BB versus gravity on falling body.

Man and projectile are both falling under gravity . I'd say it comes down to which hits the ground first. I'll stick with Pythagorean example for now .

Hi guest, In #1 I used Pythagorus to resolve the velocity into horizontal and vertical components. I then combined the equations of motion to prove mathematically what you are describing verbally, Cheers, Davo

You are all wrong! If you are the type to sit around and try to solve this type problem, you have never learned to shoot and could not hit the rascal if he was right beside you...

I agree with your comment. I couldn't hit the broad side of a barn from 5 metres let alone a bird stealing tin foil at 100 metres! (Who said the burglar was a human?)

good reply but I don't think any of you have ever shot a BB gun in your life . the trajectory and velocity and BB coeffiecnt don't add up to a target hit sorry,go out back and play with a BB gun .

Hi Guest, I agree with you in a sense, because the equations and theories we use to describe physical phenomenon are only useful if we can actually test them. But this website posts an entirely different question every week and the idea is not that we become an expert in rolling particles in cones one week, sailing via fan forced air the next week and BB experts the following week. We are engineers and at uni we didn't actually DO an experiment for everything we learnt. We learnt powerful principles that apply to an amazing range of physical problems. If it was our job to be BB engineers (I for one am not) then we'd become experts in them, but we hopefully all understand things like gravity and wind resistance and can have a fruitful discussion about something that we are not experts in. I personally enjoy contributing to the discussions because they help to keep the cobwebs out of my brain. I don't have a BB gun to play with (and by the time I find one it will be next week and I'll have to buy a trebuchet or a satelite or something) but if you would take the time to explain what "trajectory and BB coefficient" do add up to I would be interested to know.

I'm not trying to be rude, I just think your comment leads to a branch of the discussion that dies prematurely even though you are probably right. Cheers, Davo

Only in America is it considered good citizenship to shoot the man first then question his motives afterwards.

Then how come the first answer comes from Australia?

Time shift.

Interestingly or not , he is 'hit' not 'shot'

Because Australia was civilised by burglars!

Australia was not entirely civilised by burglars, most of us were the common people who were in the wrong place at the wrong time. Steal a loaf of bread. Get sent to Aussieland. Minor crimes=transportation. Major crimes=execution. Anyway a fair few of our citizens are people from across the globe.

How much for that TV, I swiped out of your house last week?

I just chucked it into me tuckerbag and went a'waltzing to my dear Matilda.

You are a coward trying to take a cheap shot at America, just because you probably don't like the President. It is clearly stated in the problem that the said jumper is a burgler. We have no information about how it is known that the man is a burgler, but he is! The police are probably the ones who chased him onto the roof!

I'm an American and proud to be one

"...shoot the man first then question his motives afterwards..."

John Wayne is reported to have said: "Somebody told me to shoot first and ask questions later. I wanted to ask him why, but I had to shoot him."

A BB gun or .177" air rifle as we call them will have a trajectory drop over 100 metres of heaps, if he falls off the building at the instant the trigger is pulled it will take about 2 secs to splat the ground & about 2 + secs for the BB or slug to cover the 100 metres, also any firearm has a velocity at the muzzle & a lesser velocity at 100 metres, unless you "held over " a distance equal to the drop over 100 metres you would miss even if he didn't fall off the building, any way I'd say NO, he would have hit he ground before the slug had covered the distance. Anyway I'd have used a .308 W with a 4-12 x scope & the answer would be a definite YES.!!!

Let us just suppose that you are standing in the window of your own building and that building is taller and you are at the top most floor of that building at an angular distance of one hundred meters. Then you will be shooting at a downward angle reducing the amount of deflective drop due to gravity and possibly lessening the effect of friction speed reduction on the bb itself. Then how quick could the projectile get to the intended victim, provided we are not in an alternate reality where people live in a vacuum and gravity does not affect linear projectiles?

I think you've mostly got it. The problem does not state that you are standing on the ground, only that you're 100 meters away and, in order to actually see the roof you would have to be above it.

Hi double j b, What is an angular distance? The higher you are above the burgular, the less the angular deviation will be (in the extreme if you are above him it will be zero), but the vertical deviation is a consequence of gravity and will be the same over equal periods time.

All this is still ignoring wind resistance which we know is unrealistic but at least it lets us isolate one variable at time for simplicity sake, so please don't shoot me. Davo

As he starts falling at least 1.7-seconds before the BB can reach him, I'd agree with those who say something else got to him first, so he wouldn't be anywhere near where I was aiming by the time my efforts could reach him (even if my BB gun sights were calibrated to take account of both the distance and the inclination of the trajectory, and I could aim accurately). So, no - and no need for all that algebra

Fyz

I've seen the same example in my 12th grade physics that is referenced in post #5.

I don't have the equations, but I do remember the slow moving projectile dropping vertically to strike the falling object ..... every single time. It didn't matter if you blew hard or soft, if the marble got far enough to strike the dropped object, then it hit it squarely every time.

Try it at home (probably outside if possible) you need a metal tube, marble, wire, battery, electro-magnet and a tin can. You set the can within range, connect the battery to the tube and the electro-magnet and close the curcuit with the electro-magent wire run to the end of the tube. Have the "end of the tube wire" touching the exit and allow it to move (and break the circuit) when the marble is launched.

Aim the tube directly at the can and puff the marble at it.

It will hit the tin can every time.

Please write back and tell us of your success stories........

I think you mean post #4....

The question says that you aim at the burglar, not that the barrel points directly at him. If the latter, you will hit him unless the building gets in the way or air resistance introduces excessive error. Again, you only need to check that he doesn't hit the ground first - otherwise, there's no need for algebra, just Newtonian relativity.

yes, ..... but if you are aiming to hit him (and he does not fall the instant you let fly), you must point the BB gun well above his body to "get" him, knowing that the projectile will drop during flight.

Therefore, if you aim at him (and he's being a good chap by playing along and falling from the 20 meter building) the projectile will drop (matching his pace) and you will strike him exactly where you aimed at (using a laser sighted BB gun if you'd like).

Does Gravity not affect the moving projectile for the same time it affects the falling man ? In the 1.7 odd seconds that the bullet takes to get there he has not yet hit the deck . The downward component of the bullets path keeps pace with the mans . I mean both objects have a downward acceleration of 9.81 ms^-2 , but the bullet keeps closing on the man at 60 m/s at an 11.3^o angle .

My interpretation was that you aimed to hit the man, not that you pointed the barrel of the BB gun at him. In the latter case, you would hit him if you lived in a vacuum (that's a simple problem of relative motion). However, as it is air that you are both (presumably) breathing, the BB would need to be inches in diameter for there to be any chance of a collision, and I don't know where you'd find a gun to fire it.

Fyz

No.... your BB had insufficient muzzle velocity...And to make matters worse ,the burglar thought you had a real rifle, pulled out his pistol, and shot you....Can I have your BB gun if you don't make it??

Anyone who's every gone duck hunting know a 12 gage shotgun is only good for about 60 meters, a BB gun has NO CHANCE to travel 100 Meters. so...

NO! you would not hit him, nor scare him, but he knew it was a BB gun, so he was laughing so hard he fell to his death. , bad news is, he landed on your car.

You should use the numbers given in the problem. The problem say that you BB gun (a special one, of course!) shoots at an initial velocity of 60 m/s.

In order to solve the problem you must use the specifications given.

Abe

hi Davo...it was not the bb or fall that got him, it was the 30,000cfm blower that landed on him.

That is true about the 12 gage shotgun is only "Good for about 60 Yards", the distance for an effective kill without much chance of wounding for a well placed shot varies between 35 and 60 yards depeding on the shot size, powder charge, and bird being shot at (size, how thick its feathers are such as a goose vs a dove).

However, the bb's can travel past 100 meters (they just might not do much once they get there). Also some air rifles (BB guns) can have muzzel velocities up to 1250 feet per second or 381 meters per second so the time it takes to cover the 100 m(horizontal distance) can vary greaty depending on the bb gun. But for some it could cover the distance in about .3 seconds.

When I was young I had a BB gun, and got good enough shooting it that i could hit a soda can at 100 yards (just about 100 meters) just about every shot, although be it, I had to aim about a meter high to accomplish it with the gun sighted in at about 10 meters. Basically I'll attest that it is possible for the BB to travel 100 meters horizontally, although the amount of kenetic energy left at that point wouldn't be enough to cause much damage. Its not the fall or the BB that you have to worry about, its the ground!

I should have read the problem, this is a pokey BB gun at 60m/s.

Remember that this BB gun does not kill. It just captures!

Abe

No. The effective range of a .22 is about 100m! That is, if you expect to reliably hit anything.

A BB won't have a ghost of a chance at that muzzle velocity to even reach the target. There simply isn't enough energy for the mass and cross sectional area of the BB to get there.

However, that is not what the question really is after. The answer has to do with gravity and the rate that an object falls. In a simplistic way, look at it this way. If you fire an object out of a muzzle horizontally and at the same time drop a rock at the same height as the muzzle, both the bullet and rock will hit the ground at the same time. I think that is the nut of this challenge, but it is poorly worded and not grounded in reality.

You don't even need the horizontal constraint.

Fyz

Precisely , and the bullet gets to him before he can reach the deck . As far as he's bothered the bullet is closing at 60 m/s from a constant 11.3^o. He gets shot , then pancaked.

So you've got gravity, but no wind resistence?? Sorry you can't have it both ways...The wind resistence would limit the BB's range to well short of 100 meters, if you say there's no air, then you are in space and there's no gravity...If you wanted to ignore certain factors, then why didn't you say so? A 60m/s velocity is extremely low, even for an air rifle, and if there's no air, then what's propelling the BB? These challenge questions are based on the premise that you have heard them before and know what the quiz master is trying to ask, because he is unable to construct the question in a lucid fashion...As such it is prejudicial in nature, and politically incorrect, and requires corrective action immediately! Such scientific questions with no attention to detail is unacceptable...It's an affront to the sensitive yet enormous intellect that drives this civilization forward and can't be tolerated...Rise up, RISE UP I SAY!!

Surely gravity without significant air resistance is perfectly possible. However, I agree that the statement that both the burglar and I might be there at some time near the present day, and near a 20-metre high building rather limits the possibilities.

Just to correlate your voice of experience with theory:
If it's a true BB gun, the pellet diameter should be 4.57-mm. Assuming it's lead (best case), and has a CD of 0.1, then its kinetic energy would decay exponentially by a factor of 1/e every 29-metres - by 40-metres, the velocity would roughly be halved.

Fyz

A standard 6 mm BB weighs about .12 grams, 60m/s = 196.850 f/s = muzzle energy .21 joule ...Assuming a level shot...

This would give us a 1 meter drop in 26.64 meters, this would be considered the effective range of this rifle...

If on the other hand we want to ignore all the other factors and just deal with gravity... We get the man hitting the ground in 2 sec...The BB traveling 100m in 1.67 sec. Therefore with this twisted logic we would indeed hit the poor burglar an instant before he hit the pool, and realize he was a high diver practicing for his assault on the title at the World championship next week....You are in so much trouble...

Why are you in so much trouble? Because the BB you fired is impervious to the forces of nature, it's muzzle velocity remains constant no matter what...It would penetrate the man with an explosive force, continue on through the building and anything else in it's unfortunate path...Affected only by gravity it would penetrate the Earth to the core possibly cracking the mantle and ending all life as we know it....OK,,, probably not...

I agree, many (most?) of the challenges are poorly phrased. The result is some interesting discussion but that is a bonus and I doubt the intention. Unless you know the answer by intuition (or have seen that monkey + electromagnet demonstration and so recognize what is wanted) , it is easy to reach a valid alternative conclusion. I assumed also that the jump was unexpected and that in taking aim the intention would be to hit his original position and you would have accounted for drop by sighting higher. So, no.

You might hit him - but it will not be while he is still standing on the roof and it depends on your height relative to the burglar. If you are standing on a hill 100M away at an elevation of 20M off the ground then yes, you could concievably hit him since the BB and the burglar will be falling at the same rate and the BB will hit the burglar approximately 1.6667 seconds (ignoring any decelleration due to drag) after his fall starts (6.37 M above the ground) and shortly before his unfourtunate encounter with the ground - assuming that the gun is sighted to shoot completely level. If your initial elevation relative to the burglar is higher or lower than your BB will either hit the building before the burglar passes on his fall (your elevation is higher than the burglar) or after the burglar passes (your elevation is lower).

My friend,

If you have the means and the time then please try the experiment of posting #32 and #4 for yourself.

You will find that no matter what angle you start with and no matter what velocity the projectile travels at, if you aim the tube directly at the object and the projectile is allowed to cross the path of the dropped item (before hitting the ground), it will always and I repeat always hit the dropped item.... every time ..... without exception.

Guess,

You are absolutely right!

You do not have to do the experiment. Just develop the equation of motion of the problem.

Abe

No chance. Air resistance means he will hit the ground long before the BB gets there

Funny, this is a problem that is in many high school physics texts. In most, the textbook writer assumes that the student has never shot a gun, and has no understanding of how a gun might be aimed. Anyone who has shot a gun knows that the barrel of the gun does not point directly at the target -- it cannot for the obvious physics reason. In prepping a rifle to be fired, you sight it, meaning that you adjust the sights to put the bullet on target at a particular distance, compensating for trajectory shape. Any marksman (and almost any kid with a bb gun) knows that you must then aim even higher for longer distances. With a bb gun, you will need to aim much higher for a long distance, because you will have sighted it for 10 meters, not 100.

But in any case, the aiming process (both the initial sight setup, and the aiming in the field) should completely compensate for the very arc (due to gravitational acceleration) that the question relies upon for its entirely wrong standard answer.

Consider this to clear up any confusion. You "aim" at the robber. He holds still. If you are half a marksman, you will hit him. Ask any cop. You have, in your aiming process, compensated completely for the trajectory curve. If that were not the case, people would never hit still targets! To suggest that the robber must drop to be hit is nonsense.

In the ludicrous case where you align the centerline of the barrel with the robber -- only then is the question's standard answer correct. But any kid, and especially any physics student, should know better: bullets do not travel in straight lines.

Ken,

Lighten up! This is not a test of marksmanship. And the same point you are making has already been made by others. Yes, of course the case is ludicrous, most of these "challenges" usually are because they set up artificial conditions that one would not find in the "real" world. Deal with it!

PS- See my answer below.

Hi STL,

You correctly discerned that I have my undies in a twist over this. (Although my concern has to do with more general trends, such as the recent enthrallment with perpetual motion machines, free energy, etc.)

Simply "Dealing with it" is good advice, but it doesn't fit my curmudgeonly personality. See my post in response to Davo's response number 19 (which references Jorie's post 18.)

These challenge questions remind me of the "barometer question".

Hi Ken

As you will see from my responses, I feel much the same way about this sort of question and about most of the answers. My take on it: the question says (as ever) that I am the active agent. Now, my aim with any sort of pistol is almost certainly terrible (blame the last forty-plus years of abstinence from armed activity), but there's no way it would be so bad the the barrel would be pointing at the miscreant. Then there's the minor detail that, whichever way I direct the weaponlet, we are both apparently breathing - so the air resistance must be sufficient that the (presumed) burglar would go splat significantly before the charge could reach him. Finally, even if we consciously make the ridiculous simplifying assumptions - I agree the irrelevance of using algebra, as the principle of Newtonian relativity was needed before the algebraic equations could be developed.

However, having said all that, even if a real-life equivalent question were to be treated in this way by one of my staff (note subjunctive - none of them is THAT incompetent), I would at least try to keep my cool and merely explain the issues. (I'd probably suggest that they "look around" if there were several similar lapses, however). Such apparent (if not real) detachment SHOULD be easier when you have the opportunity to read your first outpourings. [Sorry to over-emphasise some words - I'd have preferred an intermediate font].

Regards

Fyz

Hi fyz:

You write: I would at least try to keep my cool and merely explain the issues. You can write that because you have some cool to keep.

I thought my first post* was pretty tame. The second one started with the tone of reinforcing Davo's and Jories' correct answers. But having read STL's own rather long posts in the interim, (which both propose the "wrong" answer and also go into rather great length in doing the math, which, to me, seems superfluous) that tone changes as the post wears on. It comes across as something of a rant, doesn't it? I've always liked STL's responses to questions, so hate to see him "going over to the dark side" , and thus the emotion comes in. (Of course, I'm exaggerating here -- I continue to think of him as a very reasonable guy.)

Such apparent (if not real) detachment SHOULD be easier when you have the opportunity to read your first outpourings.

What, writing this stuff is bad enough: you don't seriously expect me to subsequently read it, do you?

Regards, and with respect for your reasoned approach,

Ken

* (in response to post #52 -- which I thought was, itself, a little over-the-top with "... and I repeat..." and the underlining, and the certainty that the burglar would always be hit no matter what the conditions)

Yes. You got him!

It is simple <grin> ballistics. Neglecting wind, differences in terminal velocity, etc. if you point (assuming plain "iron" sights, i.e. no adjustments for "windage and elevation") and shoot, the BB will follow a ballistic path. Range, x, is 100 meters. You need to determine the time,t, to reach this range, x, and the height of impact, y. If the height of the falling man is the same or nearly the same at the same time, t, you plugged him, pardner! Since the man falls under acceleration of gravity from an initial height of 20 meters, his height at time, t, is y=20-(gt²/2), where g=9.8 m/sec.², acceleration due to gravity.

Your angle of elevation is based on the height, h, you are pointing at and the horizontal distance, again, the range, x. By trigonometry (not "triggernometry"!) the tangent of the angle is equal to the height divided by the range.

So, angle Θ= arctangent (20/100) = 11.30993247... = 11.31°

I won't go through the hairy ballistics calculations, (there is a nifty ballistics calculator at this URL), but the height, h, at 100 meters range for 60 m/s and 11.31° is:

h=5.8445603405059785= 5.845 m and time:

t=1.6996735718290636=1.70 seconds

Plugging (no pun intended) time into the falling man's height equation, y=20-(gt²/2), we get:

y=20-(9.8 x (1.7)²/2)= 5.839 m, which, as we used to say, is close enough for Government Work! (The difference is only 6 mm)

QED

(What is QED? Latin for Quod erat demonstratum, which means,"That which was to be shown", or as my high school Algebra teacher used to say, "Quite easily done!"

OK, maybe I should have used 9.81 m/sec, this would put my difference to about 20 mm, still negligible, less than one inch.

And the difference is only due to limited precision in your calculation. This is a case of relative motion, both accelerating equally under gravity, so both seeing the same perturbation from the gavity-free condition. No algebra needed - except to check that the burglar doesn't hit the ground before the ball-bearing hits him.

Fyz

Well, yeah....if you want to get intuitive about it! <grin>

Sometimes problems are intuitive, sometimes they are counter-intuitive, that is why I like to run through the calculations. We call this a "sanity check". In other words, do the numbers correlate to what I thought it would be?

I do take exception to your last statement though: "No algebra needed - except to check that the burglar doesn't hit the ground before the ball-bearing hits him."

Given the conditions stated, and neglecting for air friction (terminal velocity) differences, all facetiousness aside, how could the burglar possibly hit the ground before the BB hits him? Assuming the only difference might be the height the burglar jumps from (say if he fell out a window instead of off the roof, or climbed up a ladder on the edge of the roof), as long as the gun has sufficient angle of elevation when pointing at the burglar to achieve the range of 100 meters, as you so eloquently pointed out, "both accelerating equally under gravity", won't the BB always impact the burglar at or before the instant the burglar hits the ground?

Obviously, there is a condition where the burglar jumps from such a low height, if you point the gun at him at this low angle, the BB has insufficient arc to achieve the stated range and he will "hit the ground before the ball-bearing hits him" simply because the BB can never get to him!

Now that's intuitive!

It doesn't matter whether you consider the time the burglar hits the ground or the time the BB does; neglecting air etc., if the burglar hits the ground before the BB gets to him, the BB will hit the ground before it reaches him, and vice versa. I chose the burglar because the calculation of his time to hit the ground has one less term (vertical start velocity) than the equivalent calculation for the BB. In any event you can't escape Kris's Pythagoras calculation if you want to be certain that the ball bearing will arrive before he hits the ground.

I wouldn't describe the principle as intuitive, as it is a statement of some of the fundamental laws that the calculations are based on. However, I've no problem with sanity checks - but I feel the principle is more fundamental, so I'd be inclined to say what it is, and that the sanity check is to confirm that I'm not misapplying it.

Call me a pedant if you wish .

Fyz

If I didn't I'd wish I had used a real gun and fired it just a tad sooner. All burgulars should be shot.

Don't worry about it the MoFu just dropped 20M - Splat.

Hi All,

Sorry it took so long to get to this challenge...

I was stranded in the middle of this lake, you see and...well I had this big blower with me and...I lost power to my engine....

Ok so it took awhile to swim to shore...especially since I had to tow the dag-blamin boat....

The BB will reach the target (the alleged burglar who in the US is innocent until proven guilty and I was just havin a little fun taking pot shots with my red ryder at people jumpin from buildings today) ~.3 sec before he/she jumper (politically correct) hits the ground and makes an awful mess on the sidewalk. Seeing that I could be accused of causing the death of the alleged thief and knowing how the left leaning media would crucify me I ran to offer my assistance to the splattered remains of the crook only to find that my BB was nowhere to be found. However, in my haste to retrieve the BB from the scene of my misdeed I was found to have been caring what appeared to be a high powered rifle and as I turned to confront the hundreds of shocked onlookers Ralph, the security guard in the 20m building let loose an assail of gunfire from atop the building in my general direction missing with all his shots accept one which hit my red ryder BB gun and I dropped it where is summarily hit the ground and let loose one more BB which struck Ralph right on the tip of his nose. Now my question is....how fast was the errant BB travelling when it struck Ralph's nose?

Isn't the building in the way?