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Andre's

This presentation explores the motion of a particle on the x-axis, focusing on its acceleration, velocity, and distance. It includes a sign study to analyze the velocity function and determines whether the velocity exceeds 50.

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Andre's

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  1. Andre's Presentation

  2. A particle moves on the x-axis so that its acceleration at any time t>0 is given by a(t)= When t=1, v= , and s= . The Problem

  3. Velocity We must find the velocity in terms of t. We know that: Our next step is to solve for the ANTIDERIVATIVE

  4. so To solve for C we must use the information given to us which says v(1) = The Antiderivative The antiderivative of A(t) is V(t)

  5. Solving for C Therefore So...

  6. Part B In the next part of the question we will analyze the velocity function and answer the question... Does the numerical value of the velocity ever exceed 50? The best way to consider this question is to find the acceleration’s critical values and create a sign study.

  7. The Sign Study The only critical value in the given domain is t=2. V(t) decreasing increasing > + A(t) 0 2 We know that acceleration is positive for all values greater than 2. Since acceleration is velocity’s derivative, this tells us that for all values of t>2, velocity is increasing. We can also solve the equation.

  8. Let’s solve the equation for 50. When t = 28.416, the velocity function yields 50. Since the function is increasing at this point we know it will exceed 50. We can also see it on velocity’s graph. V(t) [0,30] by [-5,55]

  9. Distance The last step is to find the distance s from the origin at time t=2. We must find the distance formula, also known as the antiderivative of velocity.

  10. Using the given condition that s(1) = we find c to be 1, and the final distance formula is...

  11. In order to find the distance s from the origin at time t=2 we simply evaluate the distance formula at t=2. Therefore, the distance from the origin at time t=2 is .86

  12. The End

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