03年美国数学建模A题

Cardboard Stacking Type When it Comes to Crashing

Summary

This article is about a stunt person on a motorcycle will jump over an elephant and land in a pile of cardboard boxes to cushion their fall. Firstly we need to protect the stunt person, secondly, we need use relatively few cardboard boxes.

In question 1, we need figure out the size, the amount and the stacked method. First we analyze the shock to single cardboard box, and draw the motion curve of the stunt person & motorcycle. Next we apply Newton's laws of motion and law of energy conservation to build the motion model. Finally we figure out the size of the cardboard box is )(303030in ⨯⨯, the number of the cardboard boxes is 138, the costing is $690.

In question 2, on the motion curve of the stunt person & motorcycle, we find that we can stack the boxes to steps. This way not only can make the stunt person safe, but also reduce the costing. It is greatly optimizing the model. So we build the stacked step model, and we also can figure out the minimum number of the cardboard boxes is: 94127228328=⨯⨯+⨯⨯+⨯⨯, and the minimum costing of the cardboard boxes is: 470594=⨯, it can save $220.

At last, we also talk about some other stacked way, we compare the conventional stacked to the intercross stacked. First we use eight cardboard boxes to analyze the force, and then popularize to n cardboard boxes, finally we find that the shock force to intercross stacked can bear is vertical p n λ)1(- times to conventional stacked.

In question 3, we generalize to different combined weights (stunt person & motorcycle) and different jump heights, due to formula (1) and (2), we figure out the number and costing to different combined weights and jump heights. At last, we get a conclusion: when the combined weigh is invariant, the amount is increasing along with the jump height; when the jump height is invariant, the amount is increasing along with the combined weight.

Contents

Cardboard Stacking Type When it Comes to Crashing (1)

Summary (1)

1. Introduction (3)

2. Restatement of the Problem (4)

3. Analysis of the Problem (4)

4. Assumptions (5)

5. the Symbols (6)

6. Model Design and Solve (6)

Model 1 the Motion Model Based on Law of Energy Conservation (6)

Model 2 the Stacked Step Model (9)

Model 3 the Size and Amount to Different Combinations (12)

7. Sensitivity Analysis (12)

8. Further Development (13)

9. Strengths and Weaknesses of the Models (13)

1. Strengths (133)

2. Weaknesses (144)

References (155)