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Old Feb 23rd 2013, 04:47 PM   #1
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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
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Old Feb 23rd 2013, 06:04 PM   #2
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Originally Posted by toranc3 View Post
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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Old Feb 23rd 2013, 06:11 PM   #3
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Originally Posted by topsquark View Post
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.
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Old Feb 23rd 2013, 06:41 PM   #4
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Originally Posted by toranc3 View Post
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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Old Feb 23rd 2013, 06:47 PM   #5
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Originally Posted by DrPhil View Post
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?
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Old Feb 23rd 2013, 07:39 PM   #6
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Originally Posted by toranc3 View Post
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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Old Feb 23rd 2013, 08:01 PM   #7
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Originally Posted by DrPhil View Post
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.
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Old Feb 24th 2013, 08:00 AM   #8
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Originally Posted by toranc3 View Post
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) ~ 1000p
That is close enough!
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