pretty please with a cherry on top ?
The Cart Problem:
You are working as a scientific analyst for a manufacturing company. The research and development (R&D) division of the company is investigating the possible use of a computer controlled supply cart to deliver parts to different stations along a manufacturing line. They hope to construct a straight track along the manufacturing line and have the cart respond to requests from different stations that need new parts. The current cart design is simple. There is a motor that can cause the cart to accellerate 1 meter per second in either the forward or reverse direction. The computer has the option at any time to run the motor in the forward direction, run the motor in the reverse direction, or turn the motor off. Before the R&D division can proceed with the project, they would like for you to make some computations and report your findings. The initial requests of the R&D division appear to be a little complex, so you decide to begin by solving a simpler problem which involves actual values. This problem can be stated as follows:
Part I: Suppose the cart is 50 meters from station A when it receives a request from the station for parts. The cart is currently travelling 2 meters per second away from station A. How must the computer control the motor so that the cart returns to station A in the quickest time possible (assuming that the cart must come to a complete stop at station A to unload the parts)?
The following information might be useful. The weight of the parts is negligible compared to the weight of the cart, and R&D has determined that the deceleration of an empty cart due to friction is roughly 1/10 of the magnitude of velocity of the cart. We will refer to the value of 1/10 in this setting as the friction coefficient.
Part II: In the process of solving the problem above you begin to ask yourself a fundamental question. Namely, you ask whether friction is helping or hindering the process. Is friction making it possible for the cart to arrive more quickly, or is friction slowing the process. To answer this question, you decide to revisit the question in part I for different values of the friction coefficient and determine whether the absence of friction would help the cart arrive more quickly and whether there is an optimal friction coefficient that would cause the cart to arrive in the quickest possible time.
The Cart Problem:
You are working as a scientific analyst for a manufacturing company. The research and development (R&D) division of the company is investigating the possible use of a computer controlled supply cart to deliver parts to different stations along a manufacturing line. They hope to construct a straight track along the manufacturing line and have the cart respond to requests from different stations that need new parts. The current cart design is simple. There is a motor that can cause the cart to accellerate 1 meter per second in either the forward or reverse direction. The computer has the option at any time to run the motor in the forward direction, run the motor in the reverse direction, or turn the motor off. Before the R&D division can proceed with the project, they would like for you to make some computations and report your findings. The initial requests of the R&D division appear to be a little complex, so you decide to begin by solving a simpler problem which involves actual values. This problem can be stated as follows:
Part I: Suppose the cart is 50 meters from station A when it receives a request from the station for parts. The cart is currently travelling 2 meters per second away from station A. How must the computer control the motor so that the cart returns to station A in the quickest time possible (assuming that the cart must come to a complete stop at station A to unload the parts)?
The following information might be useful. The weight of the parts is negligible compared to the weight of the cart, and R&D has determined that the deceleration of an empty cart due to friction is roughly 1/10 of the magnitude of velocity of the cart. We will refer to the value of 1/10 in this setting as the friction coefficient.
Part II: In the process of solving the problem above you begin to ask yourself a fundamental question. Namely, you ask whether friction is helping or hindering the process. Is friction making it possible for the cart to arrive more quickly, or is friction slowing the process. To answer this question, you decide to revisit the question in part I for different values of the friction coefficient and determine whether the absence of friction would help the cart arrive more quickly and whether there is an optimal friction coefficient that would cause the cart to arrive in the quickest possible time.
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