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# Dart Trajectory

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Already done, Hazel. see post #6

I saw it, yes, and thank you.  Amazing what we can pick up reading other people's posts.  :-)

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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 litt

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 mism

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 dif

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

I do realize that. Which is why I wouldn't - and didn't - suggest such a thing.

I was talking merely about the contact.

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While on the subject of dart trajectory would it be possible for members of this forum to offer a explanation as to what is the theoretical optimum dart throwing speed.

Studies have suggested that this to be around 5.5m/s. Apparently our brain and muscle coordination can not operate with consistent accuracy when we throw at speeds greater than this.

At the fraction of a second just prior to a dart reaching its target it is falling at a faster rate than its forward rate, (I think). I believe this faster rate starts once the peak of the parabolic trajectory curve is reached, is this correct?

It would be interesting to see a scale showing the fall in mm in the last 300mm of a darts trajectory prior to hitting the target at different speeds. eg 4.5m/s - 7.5m/s in 0.5m/s increments.

I am a member of Darts Nutz Forum and have posted a thread https://www.dartsnutz.net/forum/showthread.php?tid=33917 with a link to this topic

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At the fraction of a second just prior to a dart reaching its target it is falling at a faster rate than its forward rate, (I think).

If it were falling faster than its horizontal movement then it would, by definition, have a downward trajectory greater than 45 degrees. That would not be a good throw.

(see attachment)

I believe this faster rate starts once the peak of the parabolic trajectory curve is reached, is this correct?

No, at the peak of the parabola it has zero vertical velocity.

As you release, the dart has its highest upward (positive) vertical velocity - and a constant acceleration downwards (negative), which immediately begins canceling the positive velocity.

At the top of the parabola, all its positive velocity has been canceled by the negative acceleration, and it is momentarily vertically at zero v.

(please forgive my interchanging of speed and velocity. I could be more rigorous, but I think it would be at the cost of clarity.)

Edited by DaveC426913
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While on the subject of dart trajectory would it be possible for members of this forum to offer a explanation as to what is the theoretical optimum dart throwing speed.

Studies have suggested that this to be around 5.5m/s. Apparently our brain and muscle coordination can not operate with consistent accuracy when we throw at speeds greater than this.

There is a well-known tradeoff between speed and accuracy. Even if you were to walk over to the board and stick the dart in by hand, you will find that the faster you try to do this, the less accurate your aim is.

If you use that nice calculator that you found, you can enter various throwing angles and speeds to see what combinations will hit the target height at a distance of 2 meters.

For example, we already have seen that 20.2 degrees and 5.5 m/s is perfect.

If Gray Anderson’s angle is really 45 degrees, then his speed must be around 4.5 m/s but I doubt his angle is really that high. I would guess it is closer to 30 deg so his speed should be about 4.8 m/s to 5 m/s.

These are not necessarily the optimum throwing speeds for these angles but they are the minimum throwing speeds for these angles.

But, based upon what I said earlier about the tradeoff between speed and accuracy, my opinion (and that is all it is, an opinion) is that the optimum speed should be not much higher than the minimum throwing speed for the angle that is used.

Obviously, this will differ from person to person, based on height, arm length, muscle strength, posture and probably a hundred other factors.

The most important thing in darts is what every dart player already knows, practice, practice and more practice. The idea is to make throwing darts as natural as walking; if we stopped to think about every step we make we would probably trip over our own feet.

In the interest of full disclosure, all this “advice” is coming from somebody who has only hit a 180 once or twice in his lifetime! But I have hit a lot of 174’s for some strange reason. Must be the way the ship rolls! Cheers!

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The most important thing in darts is what every dart player already knows, practice, practice and more practice. The idea is to make throwing darts as natural as walking; if we stopped to think about every step we make we would probably trip over our own feet.

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While on the subject of dart trajectory would it be possible for members of this forum to offer a explanation as to what is the theoretical optimum dart throwing speed.

Studies have suggested that this to be around 5.5m/s. Apparently our brain and muscle coordination can not operate with consistent accuracy when we throw at speeds greater than this.

At the fraction of a second just prior to a dart reaching its target it is falling at a faster rate than its forward rate, (I think). I believe this faster rate starts once the peak of the parabolic trajectory curve is reached, is this correct?

It would be interesting to see a scale showing the fall in mm in the last 300mm of a darts trajectory prior to hitting the target at different speeds. eg 4.5m/s - 7.5m/s in 0.5m/s increments.

I am a member of Darts Nutz Forum and have posted a thread https://www.dartsnutz.net/forum/showthread.php?tid=33917 with a link to this topic

I followed your link and I see a real live "physicist" has now replied citing "conservation of momentum".

That's nice but bollocks.

For momentum to be conserved, the higher the mass the lower the velocity.

Edited by OceanBreeze
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Can someone determine the effects of drag and its effect on trajectory of the two darts - 23.5g and 24.5g quoted in this thread, they both have an approximate longitudinal surface area of . 0.00046 square meters with a frontal surface area of approx. 0.00004 square meters.

I have sourced this calculator https://www.desmos.com/calculator/on4xzwtdwz which calculates the effect of drag on a projectile, there are some parameters I am unsure of however.

Using this calculator and the following inputs:

Dart Weight = 23.5g

Launch Angle = 20.2 degrees.

Launch Speed = 5.6 1m/s

Longtitudal Surface Area = 0.00046 sq. m (is this y)

Frontal Surface Area = 0.00004 sq. m (is this x)

There is a difference of .001m after the dart reaches a target 2m from the release point. I am I correct that drag has a 1mm effect on trajectory?

Edited by Zazl
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Hi Zazi. I see you are making quite a science project out of this.

And you found another nice calculator, kudos.

I plugged in the same numbers used before, angle = 20.2 deg, velocity = 5.5 m/s, mass = .024 kg,

And using standard g = 9.81 m/s^2, air density = 1.225 kg/m^3, and I used surface areas of 0.0005 m^2 for both x and y directions (which is higher than a pub dart), and a drag coefficient of 0.2 (you can use almost any drag coefficient from 0.2 to 0.4 and there is no change because the surface area is so small) I obtained 0.2 for a tranquilizer dart. A pub dart is probably better than this.

The result is that there was less than 1 mm difference between the trajectory with or without air resistance.

But, when you start getting into air resistance then you need to be talking more about the flights than about the aerodynamic properties of the dart body. It is the flights that react to the air and stabilize the dart, much more than the dart body.

But if all you are interested in is the difference between the 23.5 g dart and the 24 g dart, as long as the flights are the same, the only difference is mass and the result we obtained before remains valid. That is, the lighter dart will float higher in accord with all the known laws of physics, just as a lighter object in any fluid floats higher.

Cheers

Edited by OceanBreeze
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Thanks again OceanBreeze,

I often read when people talk about dart flights that a larger flight creates more lift,  this implies that it affects trajectory, is this true or just another myth. I understand the concept of stability with  flights and how they effect the angle that the dart arrives at the target,larger flights = greater upward angle as the dart strikes the board.

I also believe that the flights effect stability by pivoting around the darts COG once released and that this effect can be adjusted by using larger flights (force) in conjunction with different length shafts (levers) acting on the COG (fulcrum). Is this correct,simplistic or just plain wrong?

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Thanks again OceanBreeze,

I often read when people talk about dart flights that a larger flight creates more lift,  this implies that it affects trajectory, is this true or just another myth. I understand the concept of stability with  flights and how they effect the angle that the dart arrives at the target,larger flights = greater upward angle as the dart strikes the board.

I also believe that the flights effect stability by pivoting around the darts COG once released and that this effect can be adjusted by using larger flights (force) in conjunction with different length shafts (levers) acting on the COG (fulcrum). Is this correct,simplistic or just plain wrong?

I will try to answer as best I can, with the understanding that I am not presenting myself as any sort of authority on this subject. I am a marine engineer with a solid background in physics, but you would not want me on your darts team! :lol:

At the risk of making this post too long to read, we should probably first review what we have found so far: If a lighter dart is thrown with the same force as a heavier dart, the lighter dart travels at a higher speed to the board. We also saw that resistance due to air drag makes almost no difference to the flight trajectory because of the very tiny cross-sectional area presented to the apparent wind (as long as the dart isn’t thrown sideways!).

The main things that affect the trajectory are the angle the dart is thrown, the speed of the dart and the gravitational force acting to pull the dart downwards. A lighter dart will be thrown faster for a given force and the faster the dart is thrown, the less time gravity has to act downwards on it. Therefore, a lighter dart will strike the board higher, for the same angle and force, as we have seen from the math and the calculators you have found all agree on this.

However, all of that was done without any consideration given to the flights. Flights introduce drag, but in return they stabilize the flight of the dart just as you said; by swiveling the dart about the COG.

The larger the flights, the more the aerodynamic drag they introduce; and since darts do follow a projectile trajectory (do not fly straight to the target) that drag does have some effect on trajectory.

As I see it, as the dart follows the upward part of the parabola, the dart has a positive angle of attack and the flights will create positive lift making the dart fly higher during this flight sector. But, after reaching the apex the dart starts to fall, the angle of attack is negative so the “lift” would be negative, pushing the dart downwards. All in all, the two effects from lift should nearly cancel out.

The main effect of flights is in stabilizing the dart trajectory and introducing some additional drag, so the dart will fly somewhat slower than what was calculated when not considering the flights.

Smaller flights = less drag and suitable for a faster throwing speed, lower throwing angle, and lighter darts but provide less stability correction. My guess is smaller flights are for the more experienced players or those who like to fling their darts.

Larger flights = more drag, suitable for a slower throwing speed, higher throwing angle, and heavier darts and provide more stability correction. Larger flights are probably more suitable for the less experienced players or those who like to lob their darts.

While the physics of dart flight is very interesting and good fodder for a discussion (and maybe an argument or two);   the bottom line is that everyone is different and we are not robot dart throwing machines. Consider Phil Taylor, unquestionably the greatest of all time, whose darts often stick in the board with the tail pointing down. By conventional wisdom, this should be all wrong but who is going to tell Him that? Not me!

Cheers and thanks for bringing up an interesting topic on this forum.

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Thanks mate,

Just a point on lift:

As I see it, as the dart follows the upward part of the parabola, the dart has a positive angle of attack and the flights will create positive lift making the dart fly higher during this flight sector. But, after reaching the apex the dart starts to fall, the angle of attack is negative so the “lift” would be negative, pushing the dart downwards. All in all, the two effects from lift should nearly cancel out.

Would it be correct to say that the "lift" flight sector of the parabola is shorter (in time) as once released the dart is losing speed. This would make the remaining flight sector of the parabola longer (in time). Would this time difference in positive lift compared to negative lift actually make the dart drop more than it is lifted.

Edit:

After reflecting on the above this would only be the case if the launch speed was lower than 5.5 m/s. As most dart players throw at speeds a little higher, the positive  lift flight sector would then become longer (in time) than the negative lift sector when the dart hits the target.

That trajectory  calculator is proving to be very valuable now that I have a grasp of how it works and what the inputs mean.

The only explanation I can come up with for the way Phil Taylor lands his darts in the board is that the trajectory is still in the positive flight sector and the dart is rising when it hits the board. I doubt that is the case as he would have to throw at a speed greater than 7.5 m/s. Smaller flights will mirror this effect yet the man seems to use standard flights. Would be great to know his throwing speed. There must be an answer!!!

Edited by Zazl
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The only explanation I can come up with for the way Phil Taylor lands his darts in the board is that the trajectory is still in the positive flight sector and the dart is rising when it hits the board. I doubt that is the case as he would have to throw at a speed greater than 7.5 m/s. Smaller flights will mirror this effect yet the man seems to use standard flights. Would be great to know his throwing speed. There must be an answer!!!

There is. I found this nice Pdf illustration that explains his throw.

As you can see, while most players release the dart with the tip pointing above the curve, he releases the dart with the tip pointing down, with the COG following the curve. He is actually throwing so that the apex of the parabola is closer to the board than normal, maybe half a meter from the board. This means if the board was not there, he would be throwing a normal parabola for a three-meter throw!

Using the first calculator you found, I set the angle at 25 degrees and found for a three-meter parabolic throw the speed is 6.2 m/s. That should be close to what he is doing.

The other thing is, on the downward side, during that last half-meter of travel, the slightly negative angle of attack will correct the dart’s orientation so that the tip enters the board close to horizontal or even with the tail pointing slightly downwards.

It seems a strange way to throw darts, but nobody can argue with the results.

I might try it someday but I think I am already set in my ways and too late to change.

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I think you are spot on mate, a combination of a greater release angle (from the darts COG perspective) and a slightly higher release speed the landing becomes level or slightly tail down.

I have watched a slow motion video of his action and at the release point his dart seems to rise unexpectedly at this instance.

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I dart with a guy who likes to throw underhand.

The angle of impact is supposed to be more advantageous - essentially, the dart hits the board at the top of its arc - dead straight in.

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How accurate is he?

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