Sorry didn't see this before; might need to see a more specific problem but I feel like for that sort of thing you might need to integrate again for displacement and think about what happens when x = your given range and y = 0 or something
You need to find the max displacement with the drag at a launch angle of 45°. Then find target displacement then use: (1/MaxDisp×TargDisp)×45) This gives you the Launch angle with 45 being the max achievable displacement you just proportionally find it.
What about adding a jet force to projectile? What i see could be difficult here: jet force vector must have constant length, but it's direction matches the velocity vector. So you have to add normalization term for velocity to equation. But how do you even get analytical solution for that?
Hey, at 2:40 you talk about how difficult it is to calculate the optimum angle with drag taken into account. I’m doing a study currently where I want to find exactly this, but I’m not sure how to do it. Could anyone help?
@@jngTCS I have experimental values which I'm trying to compare with theoretical values. From the experiment I've found that an angle of roughly 40 degrees maximized the horizontal displacement but I'm trying to mathematically figure out why this is the case (by creating some sort of function) How would you go about doing this?
@@tangoseven825 if you want a solution to those equations involving exponentials as a closed-form function, I think you'll need the Lambert W function just found this one, it might help :) www.researchgate.net/figure/Angle-which-maximizes-the-range-of-the-projectile-in-a-linear-resisting-medium-as-a_fig5_233034424
You integrate velocity to get displacement x(t) and y(t), then you substitute time with the corresponding term that includes x to get the ballistic curve y(x)
Easier to write v for velocity here: what we're saying is that the derivative of v is just v itself... just multiplied by some negative constant. So we'd be asking, what kind of function is pretty much its own derivative, just scaled by some constant?
Hello! How could I find what projectile launch angle I need for a given x-distance? Namely, I would launch the projectile from the point (0,0) at a certain speed v0 with an angle theta such that the projectile passes through the point (x,0). Thank you,
Solve the integrals given in this video to get x(t), y(t) ( don't forget about the integral constant that you will solve using x(0)=0 and y(0)=0 ). Next solve x(t)=x_distance in order of t and replace t in y(t)=0 with what you calculated. The result will be a mix of trigonometric/logarithmic/exponential equation that should equal 0 (that is impossible to solve in order of the angle analytically without complex functions) Next derive the equation you got in order of angle, and apply the Newton-Raphson method using initial angle 0. Obviously it's a numerical solution but shouldn't take more then 5 iterations to get the result (you can cache the result so you don't have to recalculate for every instance).
@@RafaelFixe thanks for the reply! Now, is the quadratic drag worth considering for doing ballistic calculation. In other words, does the quadratic drag make a big difference at high speeds or could it be neglected? thanks
@@BullSight_Archery At high speeds drag is mostly quadratic for something like a bullet. However solving the differential equation x''(t)=-kx(t)|v| is pretty difficult. As to the other part of your question, since the drag force will be proportional to the square of the speed, then yes, it will make a big difference, but keep in mind that for example, a bullet is already pretty aerodynamic and it's drag coefficient will be already pretty low. If you're doing a game or simulation that involves high aerodynamic objects like bullets, save yourself the effort and use linear drag, or you'll end up doing even more numerical methods to solve quadratic drag.
Hi, I know that someone already asked how to find k but is k the drag coefficient? or is it a combination of the drag coefficient, medium density, and surface area of the object and would you know of a name for K that I could search in order to be able to calculate k. thank you.
This one might be helpful: en.wikipedia.org/wiki/Drag_equation From what I can see our k is a combination of those things you mentioned (drag coefficient does feature there) and it's more quadratic drag; I'm not aware of an exact name for the k. My background is actually in (pure!) maths and I'm currently teaching high school so that's why we're less concerned with the exact physical parameters that might be in play and more seeing simple applications of vectors and differential equations etc. we study. Once we venture into the land of tertiary physics I probably need to tap out!
@@jngTCS @jng ah okay. I'm a lower 6th a student and haven't done anything related to non constant acceleration but am trying to learn it in order to write a projectile motion programme for a computer science project. Solving your equations for displacement would give me exactly what I need once I work out what k means. it was a very good video 👍
it's just a constant that depends on the physical situation - the material and shape of the object as well as the medium it's travelling through (imagine throwing a ball through the air vs through honey, for example)
When air resistance isn't neglected I have a contrasting tendency to neglect my work but this video is spectacular
Great explain! Please, could you write the equation to find the angle of launch for a known range?
This please^^^^
Sorry didn't see this before; might need to see a more specific problem but I feel like for that sort of thing you might need to integrate again for displacement and think about what happens when x = your given range and y = 0 or something
You need to find the max displacement with the drag at a launch angle of 45°.
Then find target displacement then use: (1/MaxDisp×TargDisp)×45)
This gives you the Launch angle with 45 being the max achievable displacement you just proportionally find it.
Most comprehensive explanation. Thanks.
Great video and explanation! I'm amazed at how quickly I've forgotten calculus
Hi, if you can reply immediately i need clarification please!
Great animations and clear explanations
Thank you so much. This was what I was looking for.
thnx, very clear and helpfull, now time to solve it using Eulers Method
and use it for a simulation
hi, can you make a video about calculating that angle that achieves the furthest range with air resistance
Great video! Helped a ton!
What about adding a jet force to projectile? What i see could be difficult here: jet force vector must have constant length, but it's direction matches the velocity vector. So you have to add normalization term for velocity to equation. But how do you even get analytical solution for that?
3:44 is the purple arrow itself acceleration then? whereas the green is velocity?
yes it is the acceleration due to air resistance (but not the total acceleration - note we already wrote down gravity before)
@@jngTCS Thanks - was just clarifying. Your videos are a godsend.
That was so great, thank you!!
gem of a video
nead to know the time of fligth and the final distance of motion
Hey, at 2:40 you talk about how difficult it is to calculate the optimum angle with drag taken into account. I’m doing a study currently where I want to find exactly this, but I’m not sure how to do it. Could anyone help?
Most methods I've seen end up being numerical in some way, - I guess it depends on to what extent / generality you need an exact solution!
@@jngTCS I have experimental values which I'm trying to compare with theoretical values. From the experiment I've found that an angle of roughly 40 degrees maximized the horizontal displacement but I'm trying to mathematically figure out why this is the case (by creating some sort of function) How would you go about doing this?
@@tangoseven825 if you want a solution to those equations involving exponentials as a closed-form function, I think you'll need the Lambert W function
just found this one, it might help :) www.researchgate.net/figure/Angle-which-maximizes-the-range-of-the-projectile-in-a-linear-resisting-medium-as-a_fig5_233034424
@@jngTCS Thank you! Will take a look
How would I find k for a projectile such as a tennis ball or football?
You can either be given some points in the curve or you can obtain it experimentally and then solve for this equations
do you have an answer for the integrals!
want to see if i got it correct
@@jonathansilberman1364 It should be something like
x = ucos(θ)/ k (1 - e^(-kt))
y = (ku sin(θ) + g)/k^2 (1 - e^(-kt)) - gt/k
hectic video ty big boss
Could you please explain how to get the value of k here?
What if resistance varies with angle with vertical
How can we find the displacement with the drag included?
I feel that is really important to solve for when making a ballistic object
You integrate velocity to get displacement x(t) and y(t), then you substitute time with the corresponding term that includes x to get the ballistic curve y(x)
Question: Why does the derivative of acceleration, from being a linearly negative function, turned into e^kt?
Easier to write v for velocity here: what we're saying is that the derivative of v is just v itself... just multiplied by some negative constant.
So we'd be asking, what kind of function is pretty much its own derivative, just scaled by some constant?
Hello! How could I find what projectile launch angle I need for a given x-distance? Namely, I would launch the projectile from the point (0,0) at a certain speed v0 with an angle theta such that the projectile passes through the point (x,0).
Thank you,
Solve the integrals given in this video to get x(t), y(t) ( don't forget about the integral constant that you will solve using x(0)=0 and y(0)=0 ).
Next solve x(t)=x_distance in order of t and replace t in y(t)=0 with what you calculated.
The result will be a mix of trigonometric/logarithmic/exponential equation that should equal 0 (that is impossible to solve in order of the angle analytically without complex functions)
Next derive the equation you got in order of angle, and apply the Newton-Raphson method using initial angle 0. Obviously it's a numerical solution but shouldn't take more then 5 iterations to get the result (you can cache the result so you don't have to recalculate for every instance).
@@RafaelFixe thanks for the reply! Now, is the quadratic drag worth considering for doing ballistic calculation. In other words, does the quadratic drag make a big difference at high speeds or could it be neglected?
thanks
@@BullSight_Archery At high speeds drag is mostly quadratic for something like a bullet. However solving the differential equation x''(t)=-kx(t)|v| is pretty difficult.
As to the other part of your question, since the drag force will be proportional to the square of the speed, then yes, it will make a big difference, but keep in mind that for example, a bullet is already pretty aerodynamic and it's drag coefficient will be already pretty low. If you're doing a game or simulation that involves high aerodynamic objects like bullets, save yourself the effort and use linear drag, or you'll end up doing even more numerical methods to solve quadratic drag.
Does anyone know what is A at 5:46 is? I'm kinda lost from there, thank you in advance!
It's any constant - any number could go there and the derivative would still satisfy the equation in the line above
Which softwares are used for this type of videos?
Manim :)
@@jngTCS Thanks
Hi, I know that someone already asked how to find k but is k the drag coefficient? or is it a combination of the drag coefficient, medium density, and surface area of the object and would you know of a name for K that I could search in order to be able to calculate k. thank you.
This one might be helpful: en.wikipedia.org/wiki/Drag_equation
From what I can see our k is a combination of those things you mentioned (drag coefficient does feature there) and it's more quadratic drag; I'm not aware of an exact name for the k.
My background is actually in (pure!) maths and I'm currently teaching high school so that's why we're less concerned with the exact physical parameters that might be in play and more seeing simple applications of vectors and differential equations etc. we study. Once we venture into the land of tertiary physics I probably need to tap out!
@@jngTCS @jng ah okay. I'm a lower 6th a student and haven't done anything related to non constant acceleration but am trying to learn it in order to write a projectile motion programme for a computer science project. Solving your equations for displacement would give me exactly what I need once I work out what k means. it was a very good video 👍
@@TeraGamerUK did you end up finding it?
@@martinlonegro6486 im afraid not. I completely switched what I was doing due to how annoying these equations are.
Could it be the Drag force?
thanks a lot!
hey, how do u work out what k is? or is a set value?
it's just a constant that depends on the physical situation - the material and shape of the object as well as the medium it's travelling through (imagine throwing a ball through the air vs through honey, for example)
oh okay thanks so much