The Brachistochrone

vx graph vy graph KE graph x graph y graph PE graph E tot graph
Value for radius of brachistochrone circle (m) Value for g [accel due to gravity] (m/s2) Value for q0 [starting offset] [Top is q=0, bottom is q=180] (degrees) Show circle Show Track

Notes:

You might like to verify the property of the brachistochrone that, no matter what the starting point on the "track", it always takes the same amount of time to get to the lowest point of the track. If the ball starts at the top of the track (the location corresponding to q=0), it picks up speed and travels a large distance fairly rapidly. If it starts toward the bottom of the track (a location corresponding to some q with a value up to 180 degrees), it goes more slowly because the track is "flatter". Two balls started simultaneously from two different locations will reach the lowest point of the track at the same time!

To observe/measure this time, you can change the value for q0 and re-run the simulations. If you check the "Show Track" checkbox, it will make it easier to see exactly where the lowest point of the track is. Don't forget that you can always "step" the simulation forward or backward. Note the elapsed time when the ball is at the lowest point, and then try a different value for q0!

For your information, when you choose a q0 that is nonzero, the simulation will automatically stop once the ball reaches that same starting height on the other side of the track. Of course, in the real world, the ball would continue to oscillate back and forth.