009A Sample Midterm 1, Problem 5

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The displacement from equilibrium of an object in harmonic motion on the end of a spring is:

where    is measured in feet and    is the time in seconds.

Determine the position and velocity of the object when  


Foundations:  
What is the relationship between the position    and the velocity    of an object?
       


Solution:

Step 1:  
To find the position of the object at  
we need to plug    into the equation  
Thus, we have
       
Step 2:  
Now, to find the velocity function, we need to take the derivative of the position function.
Thus, we have
       
Therefore, the velocity of the object at time    is
       


Final Answer:  
        position is  
        velocity is  

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