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.)
- Create a function that calculates the horizontal distance as a function of initial height, launch height, and launch angle.
- Calculate the horizontal distance the ball will land from the end of the ramp.
- Solve for the angle that will optimize the horizontal distance.
- 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.
- Use the Chart Component to depict how the horizontal landing distance changes as a function of angle.
- 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.
So i get
for the first iteration . Used Moment of inertia of hollow sphere .
Anyone can confirm or dissent before I go further?
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
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.
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
Ignore rotational kinetic energy. Prime 9. No Chart Component but x-y plot. Without units to solve.
