Calculating the velocity of an object down a varied slope inclined plane

[dougk]

An object starts at a height of 5m. The initial incline is 63.4 deg. After moving 2m in the x direction, the incline decreases to 7.125 deg and the motion continues for an additional 8m in the x direction. What is the total time for the object to complete the path?

Homework Equations

The initial move downward is 4m in the y and 2m in the x direction. The second move is 1m in the y and 8m in the x direction.

The Attempt at a Solution

Tried to solve it separating the moves and adding the time together.

 

[mfb]

Tried to solve it separating the moves and adding the time together.

That is a good approach. Where is the problem?

 

[dougk]

We solved for the time in the x direction and the y direction to make sure they matched. For the first leg, they did, for the second leg they didn't.

I took my kids to the math musuem in the city last week and we saw an exhibit that highlighted the fastest way down a gravitational field. We're trying to work out a simpler method by using segments.

Thanks,

Doug

 

[haruspex]

We solved for the time in the x direction and the y direction to make sure they matched. For the first leg, they did, for the second leg they didn't.

I took my kids to the math musuem in the city last week and we saw an exhibit that highlighted the fastest way down a gravitational field. We're trying to work out a simpler method by using segments.

Thanks,

Doug

Please post your working (or there's no way to tell where you went wrong).

 

[mfb]

I took my kids to the math musuem in the city last week and we saw an exhibit that highlighted the fastest way down a gravitational field. We're trying to work out a simpler method by using segments.

No finite number of segments will give the optimum. Adding more and more segments in the right way will always improve the time - with the ideal curve as limiting case.