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Submitted URL: https://app.demandgen.ptc.com/e/er?utm_campaign=Mathcad_Newsletter_Mathcad%20Minute%20MAY%20Optimal_EM%201_FY23Q3_Multi%20Asse...
Effective URL: https://community.ptc.com/t5/Mathcad/Mathcad-Community-Challenge-May-2023-Optimize-Trajectory-for/m-p/872754?src=img1&utm_...
Submission: On May 14 via api from US — Scanned from CA

Form analysis 2 forms found in the DOM

Name: form_1ab02d756c123 — POST https://community.ptc.com/t5/forums/v5/forumtopicpage.searchformv32.form.form

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            <a class="lia-link-navigation" id="postMessageLink_1ab02d7bb8da0" href="/t5/forums/postpage/board-id/PTCMathcad?message-subject=#{subject}">post question</a>
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      <span class="lia-hidden-aria-visibile" id="autocompleteInstructionsText_1ab02d756c123_0" role="alert"></span><input placeholder="Search the PTC Community" aria-label="Search" title="Search"
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      <div class="lia-autocomplete-container" style="display: none; position: absolute;">
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          <div class="lia-component-search-action-autocomplete-post-message"> didn't see an answer to your question?
            <a class="lia-link-navigation" id="postMessageLink_1ab02d7ef7922" href="/t5/forums/postpage/board-id/PTCMathcad?message-subject=#{subject}">post question</a>
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            <a class="lia-link-navigation" id="postMessageLink_1ab02d85018c6" href="/t5/forums/postpage/board-id/PTCMathcad?message-subject=#{subject}">post question</a>
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Name: form_1ab02d8842fce — POST https://community.ptc.com/t5/forums/v5/forumtopicpage.searchformv32.form.form

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MATHCAD COMMUNITY CHALLENGE MAY 2023 - OPTIMIZE TRAJECTORY FOR MAXIMUM
HORIZONTAL DISTANCE


DaveMartin
16-Pearl
‎May 01, 2023 04:13 PM
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‎May 01, 2023 04:13 PM
MATHCAD COMMUNITY CHALLENGE MAY 2023 - OPTIMIZE TRAJECTORY FOR MAXIMUM
HORIZONTAL DISTANCE

A ball with a mass of 1 kilogram is at the top of a frictionless ramp 10 meters
above the ground. The ball rolls down the incline and launches from a height of
2 meters and an angle of 30 degrees above the ground.



 

(This picture was created in Creo.)

 1. Create a function that calculates the horizontal distance as a function of
    initial height, launch height, and launch angle.
 2. Calculate the horizontal distance the ball will land from the end of the
    ramp.
 3. Solve for the angle that will optimize the horizontal distance.
 4. How will the horizontal distance change if this were performed on the Moon
    instead of on the Earth’s surface? Assume the acceleration due to gravity on
    the Moon’s surface is 1/6 that of Earth.
 5. Use the Chart Component to depict how the horizontal landing distance
    changes as a function of angle.
 6. Use a 3D Plot to show how the horizontal landing distance changes as a
    function of ramp height and launch angle. Assume the ball starts at a height
    of 10 meters.

The worksheet should contain sufficient documentation to stand on its own;
someone unfamiliar with the initial problem should be able to understand what is
being calculated.

Find the Mathcad Community Challenge Guidelines here!

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com

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24 REPLIES 24
Fred_Kohlhepp
23-Emerald I
(To:DaveMartin)
‎May 02, 2023 08:22 AM
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‎May 02, 2023 08:22 AM




Can we assume that the ramp angle (from 10 meters) is also 30 degrees? 
(Otherwise the horizontal velocity requested in #1 is not fixed.  Or do you want
that discussed too?


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DaveMartin
16-Pearl
(To:Fred_Kohlhepp)
‎May 02, 2023 10:24 AM
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‎May 02, 2023 10:24 AM




The function should calculate horizontal distance, not velocity.

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com

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Werner_E
24-Ruby V
(To:DaveMartin)
‎May 02, 2023 10:30 AM
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‎May 02, 2023 10:30 AM




> --------------------------------------------------------------------------------
> 
> @DaveMartin wrote:
> 
> 
> The function should calculate horizontal distance, not velocity.
> 
> --------------------------------------------------------------------------------

Yes, that is clear, but the distance will probably depend on the speed reached.
And this still does not answer the question whether the two legs of the ramp
actually enclose a right angle, as one could assume on the basis of the drawing.

So is the angle of the longer ramp 60 degree to the horizontal plane or should
it be considered variable?


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DaveMartin
16-Pearl
(To:Werner_E)
‎May 02, 2023 10:43 AM
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‎May 02, 2023 10:43 AM




Here are a couple hints: potential energy gets converted to kinetic energy. The
angle between the two legs of the ramp does not matter. I almost threw in the
horizontal length of the ramp as a red herring, but I thought that might get
people too caught up in the angles of the ramp. And even though I drew the angle
between the two legs of the ramp at 90 degrees, it actually doesn't factor into
the answer.

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com

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Werner_E
24-Ruby V
(To:DaveMartin)
‎May 02, 2023 02:30 PM
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‎May 02, 2023 02:30 PM




OK, but when the ball reaches zero height and is supposed to switch on to the
30° ramp, a part of the energy would be transformed into deformation and heat
(even when you assume frictionless motion). How much depends on the angle of the
first longer ramp.

But I assume that you want us to neglect this as well and also assume a
non-elastic ball with a diameter of zero so also the geometry where the two
ramps meet is not of importance, right?


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DaveMartin
16-Pearl
(To:Werner_E)
‎May 02, 2023 08:51 PM
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‎May 02, 2023 08:51 PM




Yes, please ignore deformation, heat, and diameter/volume of the ball.
Essentially it's a trajectory problem where the initial velocity comes from
potential energy.

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com

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ppal
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(To:Fred_Kohlhepp)
‎May 02, 2023 10:46 PM
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‎May 02, 2023 10:46 PM




My Attempt



 

May PTC Challenge.mcdx

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ppal
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(To:DaveMartin)
‎May 02, 2023 08:15 PM
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‎May 02, 2023 08:15 PM




So i get 



for the first iteration . Used Moment of inertia of hollow sphere .

 

Anyone can confirm or dissent before I go further?


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Werner_E
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(To:ppal)
‎May 02, 2023 09:57 PM
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‎May 02, 2023 09:57 PM




I have not spent much time on it so I may had made mistakes,  but I ended up at
approx, 17 meters



 

The desired function would be



 

or



 

 

 

 


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ppal
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(To:DaveMartin)
‎May 02, 2023 10:38 PM
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‎May 02, 2023 10:38 PM




Comments on my approach - Prime 9 attached

 

 

May PTC Challenge.mcdx

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Werner_E
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‎May 03, 2023 05:53 PM
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‎May 03, 2023 05:53 PM




Since the size of the sphere is not specified, I assume that it is simplified to
be a point mass and things like rotational energy should not matter.
@DaveMartin  could go into more detail about this.

EDIT: Ahh, I just noticed that he already wrote that the diameter of the ball
should be ignored as well. So I guess that we should also assume that all
kinetic energy is in translation and none in rotation. The ball is sliding down
and up the ramp and not rolling, hmm...

 


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DaveMartin
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(To:Werner_E)
‎May 03, 2023 06:26 PM
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‎May 03, 2023 06:26 PM




Yes, please consider this a point mass. This problem was inspired by a high
school physics problem cited by NASA in the 1960s. My intent was for people to
consider the potential energy for the object at the top of the ramp and convert
that into the kinetic energy as it comes off the ramp. That will give you the
velocity of the object, and then you can plug that into trajectory equations
like:

https://www.physicsgoeasy.com/maximum-range-of-projectile-formula/?utm_content=expand_article

For optimizing the distance in step 3, people should have an intuitive sense
that the angle would be 45 degrees. I wanted to see how people would calculate
that in Mathcad.

The main point was to get to charts and 3D plots in steps 5 and 6.

I should have probably used the word "slides" instead of "rolls" in the
explanation of the problem. I will see if I can edit that. You can ignore
friction, air resistance, rotational inertia, heat dissipation, or any other
effects. I wasn't looking for anything beyond potential energy going into
kinetic energy.

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com


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Werner_E
24-Ruby V
(To:DaveMartin)
‎May 03, 2023 06:56 PM
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‎May 03, 2023 06:56 PM




> --------------------------------------------------------------------------------
> 
> For optimizing the distance in step 3, people should have an intuitive sense
> that the angle would be 45 degrees.

Yeah, thats also what many may remember from school but it ignores the 2m ramp
height above floor



 



 

> --------------------------------------------------------------------------------
> 
> I should have probably used the word "slides" instead of "rolls" in the
> explanation of the problem.
> 
> --------------------------------------------------------------------------------

Would be unusual for a ball to be sliding instead of rolling So you would have
to use a different point object which slides.

But #1 needs clarification. Which "horizontal distance"? Probably you mean the
horizontal distance from the end of the launch ramp depending on time
(additionally to the asked for initial height, launch height, and launch angle).
So a function with four arguments, not just three. But without friction, air
resistance, etc. its just a simple function linear in time.
If you meant in #1 a function for the landing distance,  #2 would simply mean to
evaluate that function for the given input values.

 

 


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DaveMartin
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(To:Werner_E)
‎May 03, 2023 10:42 PM
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‎May 03, 2023 10:42 PM




Yes, it's not exactly 45 degrees as I altered the original problem. I changed it
slightly so that the ground is not level with the end of the ramp, as shown in
this problem:

https://homework.study.com/explanation/a-block-slides-down-a-frictionless-incline-that-terminates-in...

Let's please keep it simple. I wasn't going for anything tricky or complex. In
#1, I mean the horizontal distance from the end of the ramp when the ball lands.

Yes, a ball rolls, it doesn't slide. But I had a nice model for a ball in Creo
that I could use to make the image, so I did.

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com


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Werner_E
24-Ruby V
(To:DaveMartin)
‎May 03, 2023 10:14 PM
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‎May 03, 2023 10:14 PM




So anyway here is my entry - out of competition as it was created in Mathcad 15.
But this version allows a much quicker and faster workflow and is significantly
more convenient to use. Quite apart from the fact that it is also more powerful
in terms of functionality.

I briefly tried converting the file to Prime 9, but despite the fact that it
would have required a lot of reformatting to make it look anything like the
same, I also had to refuseto put up with the many shortcomings of Prime and the
converter to bang around. The well known auto-label bug has struck again - h has
become a constant, g an undefined variable without a way to change it. And
Prime's inability to deal with incomplete functions would have required a
workaround in calculating the optimal angle. I didn't even get as far as the
plots then. I quickly gave up trying to create a Prime Worksheet, very
disgusted, and instead attached a PDF Print in addition to the Mathcad file.

Have fun!

Here the pics of the two pages



 



 

 

WE_20230504_May Challenge.xmcdz

WE_20230504_May Challenge.pdf
485 KB

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DaveMartin
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(To:Werner_E)
‎May 03, 2023 11:00 PM
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‎May 03, 2023 11:00 PM




Looks like you got the trick answer to #4. The acceleration due to gravity falls
out of the equation. So it doesn't matter if you're doing it on the Earth, the
Moon, or Pluto.

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com

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Fred_Kohlhepp
23-Emerald I
(To:DaveMartin)
‎May 03, 2023 10:49 PM
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‎May 03, 2023 10:49 PM




Solution attached, Prime 4.0 Express

challenge 5_23 chgedA.mcdx

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JeffH1
13-Aquamarine
(To:DaveMartin)
‎May 05, 2023 07:23 AM
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‎May 05, 2023 07:23 AM




My submission in Prime 9.0 format.

 

Challenge May 2023 - JPH.mcdx

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ttokoro
19-Tanzanite
(To:DaveMartin)
‎May 05, 2023 07:41 PM
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‎May 05, 2023 07:41 PM




Ignore rotational kinetic energy. Prime 9. No Chart Component but x-y plot.
Without units to solve.



May2023 PTC Challenge.mcdx

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AlanStevens
16-Pearl
(To:DaveMartin)
‎May 08, 2023 08:01 AM
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‎May 08, 2023 08:01 AM




Here's my limited attempt with Prime 8 Express:



MayChallenge.mcdx

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JE_9022802
2-Guest
(To:DaveMartin)
‎May 08, 2023 09:40 AM
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‎May 08, 2023 09:40 AM




If it's a ball, why does the incline need to be frictionless?  The intent of a
ball is to have it rotate.  For that to occur, you need nominal friction.  


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DaveMartin
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(To:JE_9022802)
‎May 08, 2023 09:54 AM
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‎May 08, 2023 09:54 AM




If you want the incline to have friction, add friction. If you want to
incorporate that and rotational inertia, have at it.

I will repeat my original intent. I modified an old high school problem, where
the intent was for people to use the potential energy to find the initial
velocity to calculate a trajectory. Anything else you want to add to the problem
is up to you.

As with the previous Mathcad Challenges, the point isn't the answer you get, but
how you choose to approach the problem with the tools in Mathcad.

Dave Martin - dmartin@creowindchill.com - https://www.creowindchill.com

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JeffH1
13-Aquamarine
(To:JE_9022802)
‎May 08, 2023 11:07 PM
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‎May 08, 2023 11:07 PM




@JE_9022802 - I'm with you.  As much as 30% of the kinetic energy goes into
rotational inertia.  Without it, the problem is non-physical and sets a bad
precedent, even for high schoolers.  It is workable with the rotational
component.  Give it a go.


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ppal
17-Peridot
(To:DaveMartin)
‎May 09, 2023 11:14 PM
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‎May 09, 2023 11:14 PM




 

Prime 9 file











 



 



 



 







 

 

 

 

 

 

PP_ May PTC Challenge..mcdx

PP_ May PTC Challenge.pdf
758 KB

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