Brachistochrone Curve Problem

This experiment will demonstrate the Brachistochrone Curve in effect
Adam Kehler Kyle Hesla
Grade 8

Presentation

No video provided

Hypothesis

Test 1: If the marble goes down Ramp C then it will reach the bottom first.

T%est 2: If the heaviest marble (6g) traveling down Ramp C then it will reach the bottom first.

Research

A brachistochrone curve is the path in which an object will reach the bottom in the shortest time possible. This is a curved path instead of a straight path.
The problem was initially considered by Galileo, in 1638, and later challenged by Johann Bernoulli in 1696. The problem was eventually solved by Isaac Newton in 1697.
The Brachistochrone curve is the same shape as the tautochrone curve. Both are cycloids.

Variables

Test 1: Ramp A, Ramp B, Ramp C, 5g Marbel

Test 2: Ramp A, Ramp B, Ramp C, 3g Marbel, 5g, Marbel 6g

Procedure

I'm going to set up two tables that are full the same height and symmetrical to the ground. I'm going to start the robot which will move forward once it senses a gap, it will jump. It will jump either use a mouse trap solenoid or other objects.

Observations

 

Test 1:         LEGEND: 1st = 3 points, 2nd = 2 points, 3rd = 1 point

RAMP

Test 1

Test 2

Test 3

Test 4

Test 5

Total Points

A

3rd

2nd

3rd

3rd

3rd

6

B

1st

1st

1st

1st

2nd

14

C

2nd

3rd

2nd

2nd

1st

10

Test 2

 

 

Average Time

Ramp A

 

Average Time

Ramp B

 

Avgerage Time

Ramp C

3 GRAM MARBLE

43.8ms

40.1ms

42.1ms

5 GRAM MARBLE

49.4ms

34.8ms

39.0ms

6 GRAM MARBLE

51.6ms

33.9ms

38.4ms

Analysis

Test 1 analysis:
•The experiment began on February 28th, 2025
•The experiment ended on March 4th, 2025
•The ramp with the lowest cumulative points is Ramp A (which has a 45° straight decline). This is the slowest ramp.
•The ramp with the highest cumulative points is Ramp B (the ramp with the brachistochrone curve). This is the fastest ramp
 
Test 2 analysis:
•The experiment began on March 4, 2025, and ended on March 8, 2025.
•The 3g Marble was the fastest on Ramp A but the slowest on Ramp B and C
•The 6g Marble is the fastest on Ramp B and C but the slowest on Ramp A.
•For all the marbles, Ramp B is the fastest. Which supports Hypothesis #1.

Conclusion

Conclusion 1: Ramp B was the quickest. Therefore, my hypothesis was incorrect.

Conclusion 2: The 6g marble was the quickest. This hypothesis was partially correct. It was the quickest on Ramp B and C but slowest on Ramp A.

Application

  • Brachistochrone are useful for engineering and design (EX: roller coasters). In some cases it is used to accelerate the roller coaster vehicle to the highest possible speed in the shortest vertical drop.
  • Brachistochrone curves are also involved in some sports (EX: Downhill skiing and snowboarding). You're aiming for the fastest and quickest time to get down the hill.
  • Surfing also has brachistochrone curves involved too. Most if not all waves have brachistochrone curves related or similar (most of the time not perfect).

Sources Of Error

•One source of error is the gate on the device is opened at an angle causing one of the balls to go down a split second early than the others.
•Another one could be stopping the timer too early or too late.
•There were 3D print related imperfections on the ramps that could lead to inaccurate results.

Citations

Link 1   Link 2

Acknowledgement