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 Feb 23rd 2013, 04:47 PM #1 Member   Join Date: Jan 2013 Posts: 42 Shotgun Projectile A shotgun fires a large number of pellets upward, with some pellets traveling very nearly vertically and other as much as 1 degree from the vertical. Assume that the initial speed of the pellets is uniformly 150 m/s, and ingore air resistance. 1: Within what radius from the point of firing will the pellets land? Well the problem states that the initial speed of the pellets is uniformly 150m/s. Originally I thought that the problem had no acceleration but I did not see how I was able to complete the problem with that assumption. I know that uniform speed is the same as constant speed but I am not sure why they would say that. 1: Y=yo +voy*t + 1/2a*t^(2) 0=0 + 150m/s*sin(89) -4.905m/s^(2) *t^(2) Last edited by toranc3; Feb 27th 2013 at 09:08 PM. Reason: Messed up
Feb 23rd 2013, 06:04 PM   #2

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 Originally Posted by toranc3 A shotgun fires a large number of pellets upward, with some pellets traveling very nearly vertically and other as much as 1 degree from the vertical. Assume that the initial speed of the pellets is uniformly 150 m/s, and ingore air resistance. 1: Within what radius from the point of firing will the pellets land? Well the problem states that the initial speed of the pellets is uniformly 150m/s. Originally I thought that the problem had no acceleration but I did not see how I was able to complete the problem with that assumption. I know that uniform speed is the same as constant speed but I am not sure why they would say that. 1: Y=yo +voy*t + 1/2a*t^(2) 0=0 + 150m/s*sin(89) -4.905m/s^(2) *t^(2)
I didn't actually check the numbers but the logic is good. There is a slight correction to the last line here. The middle term should have a t in it. You can then solve for the time using the quadratic formula.

-Dan
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Feb 23rd 2013, 06:11 PM   #3
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 Originally Posted by topsquark I didn't actually check the numbers but the logic is good. There is a slight correction to the last line here. The middle term should have a t in it. You can then solve for the time using the quadratic formula. -Dan
Oh yeah I must have forgotten to put the t variable there, thanks. Do you know what they mean by uniform speed? I thought that meant zero acceleration.

Last edited by toranc3; Feb 23rd 2013 at 06:39 PM.

Feb 23rd 2013, 06:41 PM   #4
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 Originally Posted by toranc3 Oh yeah I must have forgotten to put the t variable there, thanks. Do you know what they mean by uniform speed? I thought that meants zero accleration.
It means that all the pellets have the same initial speed. To find the radius of spread, you are dealing with a particle 1° off vertical. Find when it hits the ground. Use the horizontal component of velocity to find the radius.
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Feb 23rd 2013, 06:47 PM   #5
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 Originally Posted by DrPhil It means that all the pellets have the same initial speed. To find the radius of spread, you are dealing with a particle 1° off vertical. Find when it hits the ground. Use the horizontal component of velocity to find the radius.
Thanks Doc! Could you check my work?

Feb 23rd 2013, 07:39 PM   #6
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 Originally Posted by toranc3 A shotgun fires a large number of pellets upward, with some pellets traveling very nearly vertically and other as much as 1 degree from the vertical. Assume that the initial speed of the pellets is uniformly 150 m/s, and ingore air resistance. 1: Within what radius from the point of firing will the pellets land? Well the problem states that the initial speed of the pellets is uniformly 150m/s. Originally I thought that the problem had no acceleration but I did not see how I was able to complete the problem with that assumption. I know that uniform speed is the same as constant speed but I am not sure why they would say that. 1: Y=yo +voy*t + 1/2a*t^(2) 0=0 + 150m/s*sin(89)t -4.905m/s^(2) *t^(2) t=30.5763 seconds X=xo+vox*t x=150m/s*cos(89)*(30.5783)seconds x=80.0445meters Is the first part correct?
Looks good - and you even remembered to put the UNITS on every number! Thank you!

Only quibble I would have would be that the final result should be rounded to to 80. m, since the precision of "1°" limits the precision of the result.
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Feb 23rd 2013, 08:01 PM   #7
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 Originally Posted by DrPhil Looks good - and you even remembered to put the UNITS on every number! Thank you! Only quibble I would have would be that the final result should be rounded to to 80. m, since the precision of "1°" limits the precision of the result.
Nice and thanks! Could you help me with part b? I am not sure how to do it. Just give me a few hints.

Here is part b

If there are 1000 pellets, and they fall in a uniform distribution over a circle with the radius calculated in part a, what is the probability that atleast one pellet will fall on the head of a person who fires the shotgun? Assume that his head has a radius of 0.1m.

I know how to do probabilty with a quarter but not this ha. Also what does it mean when they say uniform distribution?

Last edited by toranc3; Feb 23rd 2013 at 08:17 PM.

Feb 24th 2013, 08:00 AM   #8
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 Originally Posted by toranc3 Nice and thanks! Could you help me with part b? I am not sure how to do it. Just give me a few hints. Here is part b If there are 1000 pellets, and they fall in a uniform distribution over a circle with the radius calculated in part a, what is the probability that atleast one pellet will fall on the head of a person who fires the shotgun? Assume that his head has a radius of 0.1m. I know how to do probabilty with a quarter but not this ha. Also what does it mean when they say uniform distribution?
The first hint is a good one to remember when doing probability of "at least one." Rather than calculate P(1) + P(2) + ..., it is "always" easier to calculate P(0) and subtract that from unity:
P(at least 1) = 1 - P(none)
Remembering that rule will save you a lot of work some day. Turns out it won't make any difference this time, but some day...

Another law of probabilities you have to use us that the probability of a number of independent events is the product of the probabilities of the individual events. So if you determine the probability q that one pellet misses the head, then you can raise that probability to the power 1000 to get P(none).
P(at least 1) = 1 - P(none) = 1 - q^1000

A uniform distribution means that the probability is equal anyplace within the the boundary. If a pellet falls at random within a circle or radius 80 m, what is the probability p that it falls withing a circle of radius 0.1m? Probability of a miss is q=1-p, and you can go from there.

-OR-
Since p is so small, you could get away with the approximation that
P(at least 1) ~ P(1) ~ 1000×p
That is close enough!
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