Perhaps tossing it from lower than receiving height in order for the catch to occur closer to the apex of the parabola. But yea setting both heights equal would give that as optimum, good to see the math work out.
Thanks for sharing! If we had more time in the video, then looking to see how the height would change the velocity is a nice follow-up question. Maybe we can do that in a follow up video.
@@MathTheWorld Maybe write the velocity as a function of height and angle and then find where the gradient is 0 graphically. It would be really cool to see a 3d graph in a video of yours!
To optimize a perfect ball toss for your child, use a soft, lightweight ball that's easy to grip, choose a safe and open space, start at a close distance, and offer lots of encouragement and patience as they develop their skills.
Hah! It was so brilliant to see you work out that maths that every human instinctively knows. 45° -right in the middle. This was one of those "finding an answer vs brute force moments" This was an absolutely brilliant explanation of WHY.
Thanks! Yes, sometimes we do the math and find out when we get to the end that there is an easier/obvious solution. I didn't know it in this case though until I did the math. There are many situations where the optimal angle is not 45, but 60 or 120. degrees. Maybe I will do a video on those.
In both approaches (minimum velocity for given distance and maximum distance for given velocity) you end up with an expression containing cos(theta)sin(theta), which is 0.5*sin(2theta). So you don't even have to do any Calculus! :D To minimise the velocity, you want 0.5*sin(2theta) to be maximum, which happens when sin(2theta) is equal to 1, so at theta = 45°. Same goes for the maximum distance, there you once again want to maximise 0.5*sin(2theta). This especially helps in the first case :D
Thanks for sharing this! I actually use that strategy sometimes when I do this in class. Maybe I should have put that in the video as well. Thanks again!
Trig identities coming to the rescue! It is sometimes really useful and satisfying to turn a sin(x)cos(x) into sin(2x)/2 because sin(2x) is much easier to work with than sin(x)cos(x).
I think I've spotted an error, although it happens to have no effect on the optimal angle. In the derivation around 4:30, you write that x(t) = 3. But that's 3 feet, and the equation for y uses 9.8 meters per second squared as the gravitational acceleration. You should either use 1 meter for x(t) or use 32 feet per second squared for the acceleration. Or, to earn a sticker for your nerd helmet, treat x(t) as an unknown constant and prove that its value doesn't affect the optimal angle. 😉
Great video! Very interesting too. It feels nice to be able to understand everything you did. I guess that's what the first semester of college does haha. BTW I think there's a mistake at 5:48, instead of pi/2 it should be pi/4, right?
I'm sure there's math involved in the velocity of the mallet hitting the marimba. I remember you talking about this forever ago. It's a rare father who can show his love by doing math lol.
There's a lot of math within music itself. I'm the wrong person to ask to explain why, but something to do with how the sounds mix well at certain frequencies
Well.... I somehow feel like it would be better to throw it a little higher than 45 deg. Yes the ball would move faster, but the kid would also have more time predicting where the ball will go. Also I feel like it is easier to catch a ball that falls vertical than a ball that flyes horizontal. And a steeper angle also helps with this. But how to find out if 55 deg or 75 deg are better than 45 deg? I guess we would have to test this on thousands of children 🤔. Someone should really get on that. Just think of all the dads and older brothers that are thouing balls at kids at random angles, without knowing the optimal angle. 😂
Great points. This is where math modeling can break down. When I made the assumption that "easiest to catch" would be "least velocity" then the math leads to one answer, but changing the assumption about what would be easiest to catch leads to a possibly different answer. In math modeling, we often start with basic assumptions (which lead to basic models) and see what we can learn before moving to more complex models. I think you are right with the idea that actual data could be used to improve on our results.
@@MathTheWorld Yes the assumptions dictates the solution. But as an engineer you will always start with a simple model and build up from there. There are lots to learn from simple models. But if we say low speed makes it easier and long time makes it easier, then we have to choose how we weight speed and time and then it is pretty clear that we choose what is "optimal" by choosing the weight between speed and time. And now the problem is what is the optimal weight and no math in the world can solve that problem 😂
No, throw the ball from infinitely far down but just hard enough so that it’s vertical velocity is zero when it reaches your sons hands, since it will take infinitely long to reach your son the horizontal velocity will be zero therefore even your son could catch it ;)
This is what UA-cam recommended me after I watched few videos on optimization techniques under Artificial Neural Networks
@@abhishekkurup technically, both are doing calculus optimizations 🤷♂️.
For me it was after watching projectile motion
Perhaps tossing it from lower than receiving height in order for the catch to occur closer to the apex of the parabola.
But yea setting both heights equal would give that as optimum, good to see the math work out.
Thanks for sharing! If we had more time in the video, then looking to see how the height would change the velocity is a nice follow-up question. Maybe we can do that in a follow up video.
@@MathTheWorld Maybe write the velocity as a function of height and angle and then find where the gradient is 0 graphically. It would be really cool to see a 3d graph in a video of yours!
To optimize a perfect ball toss for your child, use a soft, lightweight ball that's easy to grip, choose a safe and open space, start at a close distance, and offer lots of encouragement and patience as they develop their skills.
Wonderful parenting advice! Probably more useful than our video.
Hah! It was so brilliant to see you work out that maths that every human instinctively knows. 45° -right in the middle.
This was one of those "finding an answer vs brute force moments"
This was an absolutely brilliant explanation of WHY.
Thanks! Yes, sometimes we do the math and find out when we get to the end that there is an easier/obvious solution. I didn't know it in this case though until I did the math. There are many situations where the optimal angle is not 45, but 60 or 120. degrees. Maybe I will do a video on those.
Math The World, awesome content bro
Thank you so much!
In both approaches (minimum velocity for given distance and maximum distance for given velocity) you end up with an expression containing cos(theta)sin(theta), which is 0.5*sin(2theta). So you don't even have to do any Calculus! :D To minimise the velocity, you want 0.5*sin(2theta) to be maximum, which happens when sin(2theta) is equal to 1, so at theta = 45°. Same goes for the maximum distance, there you once again want to maximise 0.5*sin(2theta). This especially helps in the first case :D
Thanks for sharing this! I actually use that strategy sometimes when I do this in class. Maybe I should have put that in the video as well. Thanks again!
Trig identities coming to the rescue! It is sometimes really useful and satisfying to turn a sin(x)cos(x) into sin(2x)/2 because sin(2x) is much easier to work with than sin(x)cos(x).
Extra points for the hydralisk!
Thanks! I didn't create it, but I am inspired by the person who did!
I think I've spotted an error, although it happens to have no effect on the optimal angle. In the derivation around 4:30, you write that x(t) = 3. But that's 3 feet, and the equation for y uses 9.8 meters per second squared as the gravitational acceleration. You should either use 1 meter for x(t) or use 32 feet per second squared for the acceleration.
Or, to earn a sticker for your nerd helmet, treat x(t) as an unknown constant and prove that its value doesn't affect the optimal angle. 😉
Great video! Very interesting too. It feels nice to be able to understand everything you did. I guess that's what the first semester of college does haha.
BTW I think there's a mistake at 5:48, instead of pi/2 it should be pi/4, right?
I'm sure there's math involved in the velocity of the mallet hitting the marimba.
I remember you talking about this forever ago. It's a rare father who can show his love by doing math lol.
Dang! I can't remember that. I must be getting old.
There's a lot of math within music itself. I'm the wrong person to ask to explain why, but something to do with how the sounds mix well at certain frequencies
Sin(Th)cos(Th) = ½sin(2Th), which is easier to find max or min.
Well....
I somehow feel like it would be better to throw it a little higher than 45 deg. Yes the ball would move faster, but the kid would also have more time predicting where the ball will go.
Also I feel like it is easier to catch a ball that falls vertical than a ball that flyes horizontal. And a steeper angle also helps with this.
But how to find out if 55 deg or 75 deg are better than 45 deg?
I guess we would have to test this on thousands of children 🤔. Someone should really get on that. Just think of all the dads and older brothers that are thouing balls at kids at random angles, without knowing the optimal angle. 😂
Great points. This is where math modeling can break down. When I made the assumption that "easiest to catch" would be "least velocity" then the math leads to one answer, but changing the assumption about what would be easiest to catch leads to a possibly different answer. In math modeling, we often start with basic assumptions (which lead to basic models) and see what we can learn before moving to more complex models. I think you are right with the idea that actual data could be used to improve on our results.
@@MathTheWorld Yes the assumptions dictates the solution. But as an engineer you will always start with a simple model and build up from there. There are lots to learn from simple models.
But if we say low speed makes it easier and long time makes it easier, then we have to choose how we weight speed and time and then it is pretty clear that we choose what is "optimal" by choosing the weight between speed and time.
And now the problem is what is the optimal weight and no math in the world can solve that problem 😂
No, throw the ball from infinitely far down but just hard enough so that it’s vertical velocity is zero when it reaches your sons hands, since it will take infinitely long to reach your son the horizontal velocity will be zero therefore even your son could catch it ;)
why Gravity Vector is divided by 2 ?
x=1/2gt^2
a = g thus v = t * g + v0 thus x = t^2 / 2 *g + t v0 + x0, it's because the integral of the integral of a constant is the constant multiplied by t^2
@@ziiirozone ahok, g is the second derivative of x
@@kaaristotelancien3005 exactly 👍
It's a great and exciting channel, I hope my channel will reach this success.