Classical mechanics problem

A ball of very small size is dropped vertically onto a frictionless inclined plane which makes an angle thetha with the horizontal. Initially the distance between the plane and the ball is d and the speed of the ball is zero. The trajectory of the ball consists of parabolic arcs .what will be the locus of all those focus of all those parabolic arcs . Assume that elastic collisions is taking place and no air resistance is there

Note by Azimuddin Sheikh
1 month, 4 weeks ago

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Here is a plot of a ball dropped onto a 30 degree ramp from a height of 1 meter. The key to the simulation is that upon every bounce, the component of the velocity normal to the ramp is reversed, and the component of the velocity tangential to the ramp is preserved.

The second graph is of a ball dropped onto a 15 degree ramp from a height of 5 meters

Steven Chase - 1 month, 3 weeks ago

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Sir is it possible to join the focus of all these parabolas in the figure ? Also sir can u make a horizontal straight line passing through from the initial position ? So now from this property ( which is directrix is same horizontal line for all the parabolas) isn't it possible to easily get the locus in mathematical way sir ?? Or now also its complicated ? @Steven Chase sir if yes then I will make another problem , from this next onwards. My friends told me that focus has a nice locus . They maybe wrong though.

Azimuddin Sheikh - 1 month, 3 weeks ago

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It may be possible, but my way of solving isn't useful for deriving an analytical expression. Anyway, I'm moving on from this one, and I have posted the ramp problem in the CM section.

Steven Chase - 1 month, 3 weeks ago

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@Steven Chase Yeah OK sir , I will post a new problem then.

Azimuddin Sheikh - 1 month, 3 weeks ago

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@Azimuddin Sheikh I'll do the bead problem next

Steven Chase - 1 month, 3 weeks ago

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@Steven Chase Yeah thx sir. Was waiting for that problem to be solved .

Azimuddin Sheikh - 1 month, 3 weeks ago

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@Aaghaz Mahajan can u help ??

Azimuddin Sheikh - 1 month, 3 weeks ago

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okay...........lemme try

Aaghaz Mahajan - 1 month, 3 weeks ago

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Thx for replying sir , pls share your ideas when u have tried it. @Aaghaz Mahajan sir. BTW @Steven Chase sir can u also share your ideas ??

Azimuddin Sheikh - 1 month, 3 weeks ago

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@Azimuddin Sheikh @Aaghaz Mahajan sir u got the locus ??

Azimuddin Sheikh - 1 month, 3 weeks ago

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@Azimuddin Sheikh Hey don't call me sir.....I am only 16 yrs old....XD....!! Btw, I tried your question......it is pretty weird in my opinion......Are you sure the answer has a nice form??? Because well, it seems pretty complicated to me.......I tried but am not able to find a solution..........will surely post you if anything else came up.....

Aaghaz Mahajan - 1 month, 3 weeks ago

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@Aaghaz Mahajan Yeah it will be complicated bro. It should have a nice locus I think so. can we prove that the directrix of all the parabolas is the same horizontal straight line, passing through the initial position of the ball? Also is it true that all the parabolic arcs are touching the line which is parallel with the inclined plane and passes through the initial position of the ball? @Steven Chase sir , @Aaghaz Mahajan bro ??

Azimuddin Sheikh - 1 month, 3 weeks ago

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@Azimuddin Sheikh It is easy to derive numerical results and plot the trajectory. I can’t guarantee any kind of closed form solution though. That’s why I haven’t done this one yet.

Steven Chase - 1 month, 3 weeks ago

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@Steven Chase Sir can't we be able to find closed form by looking at the focus of some of the parabolic arcs ? Is the locus not coming out to be a nice curve by plotting the trajectory and joining all focuses?

Azimuddin Sheikh - 1 month, 3 weeks ago

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@Steven Chase Yeah I agree Sir.......but numerical results are always easy (comparatively!) and they don't really give us the true essence of solving something......So, that is why I was trying to do this manually, but in vain....:(

Aaghaz Mahajan - 1 month, 3 weeks ago

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@Aaghaz Mahajan So how about we turn this into a question that can actually be solved?

Steven Chase - 1 month, 3 weeks ago

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@Steven Chase @Aaghaz Mahajan bro @Steven Chase sir isn't it is interesting (that the directrix of all the parabolas is the same horizontal straight line, passing through the initial position of the ball), the problem is made to get interesting results isn't ?? I maybe wrong about calculation part , which can be hard .[[ [[Can anyone just show by animation what would be the locus seems like , if its not nice I will change this problem ,for sure.]]]]

Azimuddin Sheikh - 1 month, 3 weeks ago

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@Azimuddin Sheikh I can make a plot of the trajectory

Steven Chase - 1 month, 3 weeks ago

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@Steven Chase Yeah pls sir , show it

Azimuddin Sheikh - 1 month, 3 weeks ago

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@Azimuddin Sheikh Ok, it'll be some time in the next day. It's almost bedtime here

Steven Chase - 1 month, 3 weeks ago

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