MATH 1311 Section 6.5.

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Presentation transcript:

MATH 1311 Section 6.5

Equilibrium or Steady State Equations: When an object is dropped from an airplane, there are two opposing forces acting upon the object: Acceleration due to gravity will cause an object to speed up as it dropped. Slowing down due to resistance from air molecules. These will eventually balance out to an steady-state, known as terminal velocity.

Why are we still alive? Our solar system is based on two opposing forces: Centripetal force is constantly pushing us out into space (think about when you are driving around a sharp turn). Gravitational attraction from the sun is constantly pulling us towards the sun. The balance is that we are in a stable orbit…..or are we?

Equations of Equilibrium: Equilibrium or Steady State Equations must have a rate of change equal to 0. For the falling object: A = g – rV = 32 – 0.1818V [You would be given these values.] Determine the value of V, assuming that this equation is at equilibrium.

Continuing with the idea of a falling object: We saw that the value of x placing this equation at equilibrium is 176.018. We know that our limiting value of the velocity is: 176.018 and our rate of change would be: e(-1/.1818)t. We also know that velocity will begin lower than our limiting value and increase to 176.018. Find the velocity function of a falling object with air resistance:

Continuing with the idea of a falling object: We saw that the value of x placing this equation at equilibrium is 176.018. We know that our limiting value of the velocity is: 176.018 and our rate of change would be: e(-1/.1818)t. We also know that velocity will begin lower than our limiting value and increase to 176.018. Find the velocity function of a falling object with air resistance: v(t) = 176.018 - ke(-1/.1818)t

Determine the value of k is the initial velocity of the object is 0:

Determine the value of k is the initial velocity of the object is 0: k = 176.018 Making the function: v(t) = 176.018 – 176.018e(-1/.1818)t = 176.018(1 - e(-1/.1818)t) How fast the object reaches terminal velocity will be determined by the atmospheric conditions (temperature, pressure, humidity, etc). https://www.desmos.com/calculator/vu72gji5dn