Dart Trajectory

[Zazl]

Could someone help with the following please:

 

I have a theoretical dart throwing machine that launches darts weighing 24 grams at a height of 1730mm to a target situated at the same height 2000mm from the release point.

The force used to propel the dart is constant and generates a launch speed of approximately 5.5 m/s and every dart hits the target exactly 1730mm high.

My question is where will  darts hit the target if they weigh 23.5 grams and 24.5 grams respectively if launched from my theoretical dart throwing machine without altering the force and angle settings used for a 24 gram dart.

I believe that the launch speed will be higher for the lighter dart and slower for the heavier dart.

 

I expect the lighter dart to hit the target higher and the heavier dart to hit lower than the 24 gram dart, I need to know the difference in mm but have no idea how to calculate this problem.

 

Please note that the surface area of the different weighted darts are identical.

[OceanBreeze]

Are you familiar with parabolic motion problems? This is what you have here.

 

If this is homework, at least make a start to trying to find a solution and then I can help you, but I need to see a little effort on your part first.

[Zazl]

Hi,

I am a 65 year old dart player, When I was younger I was very good at maths, however in the last few years my mental capacity has diminished, I guess that is the reality of the aging process.

My problem is that a set of 24gram darts I purchased has one dart that is 0.5 grams lighter and I need to know what effect it would have on hitting what I was aiming for.

When throwing these darts I can not detect the weight difference so assume that I am throwing all three with the same force, if so then the lighter dart would have to hit higher than the other two darts. I do not know how to determine the trajectory of darts thrown with the same force and release angle but have different weights.

 

Homework is a distant memory for me, this is a practical request seeking assistance from those that can.

I hope someone can help.

[OceanBreeze]

Hi,

I am a 65 year old dart player, When I was younger I was very good at maths, however in the last few years my mental capacity has diminished, I guess that is the reality of the aging process.

My problem is that a set of 24gram darts I purchased has one dart that is 0.5 grams lighter and I need to know what effect it would have on hitting what I was aiming for.

When throwing these darts I can not detect the weight difference so assume that I am throwing all three with the same force, if so then the lighter dart would have to hit higher than the other two darts. I do not know how to determine the trajectory of darts thrown with the same force and release angle but have different weights.

 

Homework is a distant memory for me, this is a practical request seeking assistance from those that can.

I hope someone can help.

 

 

Hmmmm, this seems to be quite a different question than what you originally posed. The way your original question was asked, it seemed to be very much like a typical homework question, rather than a practical matter.

Based upon your latest response, I am not sure if you are interested in the calculations, or you just want an answer?

 

Anyway, I will provide you with the equations to use in this type of problem in projectile motion. Although the darts will follow a parabolic path, the path can be broken down in X and Y parametric equations:

 

[math]Y\quad =\quad { v }_{ o }t\quad Sin\theta \quad -\quad 1/2\quad g{ t }^{ 2 }[/math]

 

And

 

[math]X\quad =\quad { v }_{ o }t\quad Cos\vartheta[/math]

 

Since there are two variables, [math]\vartheta[/math] and t, one of the variables needs to restated in terms of the other, like so

 

[math]t\quad =\quad \frac { X }{ { v }_{ o }\quad Cos\theta  }[/math]

 

By substituting that expression for t into the expression for Y and knowing that [math]{ v }_{ o }\quad =\quad 5.5\quad \frac { m }{ s }[/math] and [math]X\quad =\quad 2\quad m[/math] we can solve for [math]\theta[/math]

 

And it turns out that [math]\theta[/math] is 20.2 degrees. That is the angle the 24 gram dart must be thrown at to hit the target at the desired height of 1.73 m.

 

As you correctly guessed, the lighter dart will hit the target higher and the heavier dart will hit lower than 1.73 m.

 

I will stop here for now. Let me know if you interested in knowing how the rest of the calculation is done, or only interested in knowing the answer so I know how to proceed.

[Zazl]

Thanks for your response OceanBreeze, these calculations are like a foreign language to me, the answers only will be much appreciated. Without knowing them I have absolutely no idea what effect a mismatched set of darts has when trying to throw consistently at a very small target area. Once again thanks for your time.

[OceanBreeze]

Ok, That is what I needed to know; whether you were interested in a detailed mathematical analysis or just the answers. So, I will just work out the change in acceleration and velocity, due to the different weights, as an approximate proportionality.

 

Since F=ma we can assume any value of force to get the proportionality so I will make F = 1.

 

Then acceleration = 1/24 for the 24 gram dart or 0.04167 (not an actual acceleration)

For the 23.5 g dart, a = 0.042553 which is 1.02 times higher than for the 24 g dart

For the 24.5 g dart, a = 0.040816 which is 0.98 times lower than for the 24 g dart

 

The resulting launch velocities will also be higher & lower by approximately the same proportions

 

That is [math]{ v }_{ 0 }\quad =\quad 5.61[/math] m/s for the 23.5 g dart

 

And [math]{ v }_{ 0 }\quad =\quad 5.39[/math] m/s for the 24.5 g dart

 

Plugging those values into the expression for Y, and keeping everything else the same you see that the lighter dart will hit the target approximately 29 mm too high and the heavier dart will hit about 30 mm too low, allowing for rounding errors.

 

I would have thought it would be perfectly symmetrical but remember there is some approximation to the answer.

 

I hope that helps you adjust your throw! Incidentally, for a real challenge, try playing darts on board a ship that is being tossed about like a cork in the churning Antarctic waters! Cheers!

Edited July 27, 2018 by OceanBreeze

[Zazl]

Thanks mate, your a legend. I have enough trouble throwing darts on the level so will give throwing them on a heaving ship a big miss!

All the best.

[Zazl]

Prior to asking for answers to this problem I found this online calculator https://www.desmos.com/calculator/gjnco6mzjo

If this calculator had an additional weight setting slider ranging from 10 grams to 50 grams in .1g increments it would make my understanding of a darts trajectory complete.

Would it be possible for this calculater be configured in this way?

[DaveC426913]

This is kind of strange.

 

Since you acknowledge that you can't tell the difference in weights, it follows that you can't adjust your throwing force to compensate simply based on weight alone.

 

Is this actually about you manually throwing darts? Or something else?

 

If it's really about throwing darts, then why not do the experiment? Get a few darts of differing weights, throw them, and plot what effect you get. You'll likely notice that heavier darts fall shorter than lighter darts.

[OceanBreeze]

Prior to asking for answers to this problem I found this online calculator https://www.desmos.com/calculator/gjnco6mzjo

If this calculator had an additional weight setting slider ranging from 10 grams to 50 grams in .1g increments it would make my understanding of a darts trajectory complete.

Would it be possible for this calculater be configured in this way?

 

 

That's a nice calculator. Thanks for sharing that link.

 

I set the angle to what I calculated, 20.2 Degrees and then set the calculated velocities to what I calculated 5.61 m/s and 5.39 m/s and the graph gave the heights at the target distance of 2 meters as + 28.8 mm and - 30.01 mm which is also what I came up with, so that all checks out.

 

I don't see any need to reconfigure the calculator at all as it can be used just the way it is. What you will need to do is use the same procedure I used to get the angle and to get the velocities then the calculator will do the rest for you.

[OceanBreeze]

This is kind of strange.

 

Since you acknowledge that you can't tell the difference in weights, it follows that you can't adjust your throwing force to compensate simply based on weight alone.

 

Is this actually about you manually throwing darts? Or something else?

 

If it's really about throwing darts, then why not do the experiment? Get a few darts of differing weights, throw them, and plot what effect you get. You'll likely notice that heavier darts fall shorter than lighter darts.

 

 

Not trying to speak for him; I think he just wants to have a better understanding about the dart trajectory. I enjoy this kind of thread.

[OceanBreeze]

 

I have a theoretical dart throwing machine that launches darts weighing 24 grams at a height of 1730mm to a target situated at the same height 2000mm from the release point.

 

 

One thing that I have been wondering about is your use of 2 meters as the target distance to the release point. I am sure you know that the distance to the oche is 2.37 m.

I assume you are making allowance for the arm length beyond the oche, but I wonder how you made your estimate of that distance?

 

[Zazl]

In reply to DaveC

The darts I purchased has 1 light dart, unless I mark this dart I have no idea which one it is, When I throw these darts I may throw the first dart a little high so I compensate by aiming the next dart a little lower. If the second dart was the light dart I should land this dart at the same point as the first, so I aim the third dart which is the other heavy dart lower, it should land lower than the second dart as my muscles and brain remember the last throw and try to compensate. This scenario does not take into account any inconsistency I have in my throw which compounds the situation. I basically have no control when throwing any of these darts as the light darts trajectory influences how I throw the others.

This situation should not exist if manufacturers made a greater effort in accurately machining the darts they sell, this exercise highlights the problems caused by poorly manufactured darts. I had no idea that the light dart would land 29mm high, if I had not weighed the darts I would have been oblivious to the havoc this light dart would create with my aim of developing a consistent throw. Consistency is impossible to achieve when the darts have a built in inconsistency of 29mm.

 

In reply to OceanBreeze

I stand fairly upright when I throw due to my age, I also stand front on with my shoulders square to the oche, by standing at the oche and dropping a string line from my normal release point to the oche the distance is 2m (2 on your illustration). Most players release a lot closer as some stand side on and also lean towards the board, my body won't allow that.

 

Not sure if I will be able to duplicate your calculations and suspect many other dart players would have the same difficulty, this calculator modified as explained would be a great help to many dart players in plotting and understanding the implications of variables that determine a darts trajectory.

[DaveC426913]

In reply to DaveC

The darts I purchased has 1 light dart, unless I mark this dart I have no idea which one it is, When I throw these darts I may throw the first dart a little high so I compensate by aiming the next dart a little lower. If the second dart was the light dart I should land this dart at the same point as the first, so I aim the third dart which is the other heavy dart lower, it should land lower than the second dart as my muscles and brain remember the last throw and try to compensate. This scenario does not take into account any inconsistency I have in my throw which compounds the situation. I basically have no control when throwing any of these darts as the light darts trajectory influences how I throw the others.

This situation should not exist if manufacturers made a greater effort in accurately machining the darts they sell, this exercise highlights the problems caused by poorly manufactured darts. I had no idea that the light dart would land 29mm high, if I had not weighed the darts I would have been oblivious to the havoc this light dart would create with my aim of developing a consistent throw. Consistency is impossible to achieve when the darts have a built in inconsistency of 29mm.

 

I'm surprised to hear that darts of the same set would not have identical weights. That's a real quality issue. It's probably too late to take them back.

 

You can buy tiny weights to add to your darts. This would be a far better solution than trying to compensate. The key to accurate darts is consistency. You don't want to be changing your throw all the time.

 

 

In reply to OceanBreeze

I stand fairly upright when I throw due to my age, I also stand front on with my shoulders square to the oche, by standing at the oche and dropping a string line from my normal release point to the oche the distance is 2m (2 on your illustration). Most players release a lot closer as some stand side on and also lean towards the board, my body won't allow that.

 

Not sure if I will be able to duplicate your calculations and suspect many other dart players would have the same difficulty, this calculator modified as explained would be a great help to many dart players in plotting and understanding the implications of variables that determine a darts trajectory.

 

To my eye, the thing that most confounds amateur dart players is that they don't realize how much the arc affects their accuracy. A dart that hits the board closer to horizontal will have a much higher probability of hitting near its target than a dart that is on its way down.

 

You're off by a half inch for a horizontal throw, you're off by a half inch.

You're off by a half inch on a dart that's coming in at a 45 degree angle, and you're off by a lot more.

 

Well, that and posture. I'm astonished at how many players I see trying to throw their darts with their whole body.

Edited July 28, 2018 by DaveC426913

[hazelm]

Thanks for your response OceanBreeze, these calculations are like a foreign language to me, the answers only will be much appreciated. Without knowing them I have absolutely no idea what effect a mismatched set of darts has when trying to throw consistently at a very small target area. Once again thanks for your time.

Echo of the request, please.  :-)

 

hazel

[OceanBreeze]

 

 

 

To my eye, the thing that most confounds amateur dart players is that they don't realize how much the arc affects their accuracy. A dart that hits the board closer to horizontal will have a much higher probability of hitting near its target than a dart that is on its way down.

 

You're off by a half inch for a horizontal throw, you're off by a half inch.

You're off by a half inch on a dart that's coming in at a 45 degree angle, and you're off by a lot more.

 

 

 

 

Try telling that to Gary Anderson! His darts look like they hit at close to 45 deg and he just threw a perfect game.

 

It is impossible to throw a dart perfectly horizontal, as I am sure you must realize.

 

 

[OceanBreeze]

Echo of the request, please.  :-)

 

hazel

 

Already done, Hazel. see post #6