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1. An object with mass m moves vertically under constant gravitational force through a medium that exerts a resistance with magnitude proportional to the speed of the object. Assume that the initial velocity is vo and the initial altitude is yo- A) Find the velocity of the object as a function of t B) Find the terminal velocity (i.e., lim v(t)). C) Find y(t), where y(t) represents the altitude of the object at any time t.

1. An object with mass m moves vertically under constant gravitational force through a medium that exerts a resistance with magnitude proportional to the speed of the object. Assume that the initial velocity is vo and the initial altitude is yo- A) Find the velocity of the object as a function of t B) Find the terminal velocity (i.e., lim v(t)). C) Find y(t), where y(t) represents the altitude of the object at any time t.

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  1. Suppose an object with initial temperature ?! is placed in a sealed container, which is in turn placed in a medium with temperature ?%. Let the initial temperature of the container be ?!. Assume that the temperature of the object does not affect the temperature of the container, which in turn does not affect the temperature of the medium. (These assumptions are reasonable, for example, if the object is a cup of coffee, the container is a house, and the medium is the )

 

  1. Assuming that the container and the medium have distinct temperature decay constants ? and ?% respectively, use Newton’s law of cooling to find the temperatures ?(?) and ?(?) of the container and object at time ?.
  2. Assuming that the container and the medium have the same temperature decay constant ?, use Newton’s law of cooling to find the temperatures ?(?) and ?(?) of the container and object at time ?.
  3. Find the limiting values of temperatures ?(?)and ?(?)
1. An object with mass m moves vertically under constant gravitational force through a medium that exerts a resistance with magnitude proportional to the speed of the object. Assume that the initial velocity is vo and the initial altitude is yo. A) Find the velocity of the object as a function of t B) Find the terminal velocity (i.e., lim v(t)). t-00 C) Find y(t), where y(t) represents the altitude of the object at any time t. 2. Suppose an object with initial temperature To is placed in a sealed container, which is in turn placed in a medium with temperature Tm: Let the initial temperature of the container be So- Assume that the temperature of the object does not affect the temperature of the container, which in turn does not affect the temperature of the medium. (These assumptions are reasonable, for example, if the object is a cup of coffee, the container is a house, and the medium is the atmosphere.) A) Assuming that the container and the medium have distinct temperature decay constants k and km respectively, use Newton's law of cooling to find the temperatures S(t) and T (t) of the container and object at time t. B) Assuming that the container and the medium have the same temperature decay constant k, use Newton's law of cooling to find the temperatures S(t) and T(t) of the container and object at time t. C) Find the limiting values of temperatures S(t) and T(t)
Transcribed Image Text:1. An object with mass m moves vertically under constant gravitational force through a medium that exerts a resistance with magnitude proportional to the speed of the object. Assume that the initial velocity is vo and the initial altitude is yo. A) Find the velocity of the object as a function of t B) Find the terminal velocity (i.e., lim v(t)). t-00 C) Find y(t), where y(t) represents the altitude of the object at any time t. 2. Suppose an object with initial temperature To is placed in a sealed container, which is in turn placed in a medium with temperature Tm: Let the initial temperature of the container be So- Assume that the temperature of the object does not affect the temperature of the container, which in turn does not affect the temperature of the medium. (These assumptions are reasonable, for example, if the object is a cup of coffee, the container is a house, and the medium is the atmosphere.) A) Assuming that the container and the medium have distinct temperature decay constants k and km respectively, use Newton's law of cooling to find the temperatures S(t) and T (t) of the container and object at time t. B) Assuming that the container and the medium have the same temperature decay constant k, use Newton's law of cooling to find the temperatures S(t) and T(t) of the container and object at time t. C) Find the limiting values of temperatures S(t) and T(t)
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