Building Sand Castle Walls: Force vs. Exponential

[russell16]

I am at a beach with my family and we were building a sand castle and we wanted to build a wall around it so it would last longer against high tide. I said to my cousins that the wall should be slanted in order to decrease the force of the water instead of being flat because then the wall would take the full force and it would not be distributed as well. And i used the term exponentially because i knew it was greater than a proportional difference and i wanted it done my way. another cousin said that he thinks it wouldn't be exponentional. he does agree that would better distribute the force but thinnks it would be a proportion. I just decided for sake of arguing to keep my side and say exponential. If anyone can provide equational proof that force of water flowing on to a slanted surface is exponential or proportional it would be apprieciated.

 

[HallsofIvy]

The force hitting a wall does not depend at all upon its being slanted. What is true is that a slanted wall has a larger base than a non-slanted wall of the same amount of material and so resists that force better.

 

[sir-pinski]

First of all, it's great that you and your cousins are engaging in a discussion about the best way to build a sand castle wall. It shows that you are thinking critically and considering different perspectives.

In terms of the argument about whether the force of water on a slanted surface is exponential or proportional, it's important to first define what we mean by these terms. Exponential growth refers to a situation where the rate of change is constantly increasing, while proportional growth means that the rate of change remains constant.

In this case, the force of water on a slanted surface would actually be proportional, not exponential. This is because the force of water is directly proportional to the height of the water column, which is determined by the depth of the water and the angle of the surface.

To understand this better, let's consider two scenarios. In the first scenario, the wall is flat and the water is able to hit it with full force. In this case, the force of the water is directly proportional to the height of the water column, which is the same as the height of the wall.

In the second scenario, the wall is slanted and the water is able to hit it at an angle. In this case, the force of the water is still directly proportional to the height of the water column, but the height of the water column is now shorter because of the angle of the wall. Therefore, the force of the water is reduced, but it is still proportional to the height of the water column.

In conclusion, while building a slanted wall may help to better distribute the force of the water, it does not result in exponential growth of the force. It is important to consider both the distribution of force and the overall magnitude of the force when building a sand castle wall.