Recall that Newton’s Equation of Motion for a Body of Mass, m, is F = m(dv/dt). Assume the Initial Velocity of the Projectile is v0, and the Gravitational Acceleration Constant is g, where the Gravitational Force is Fg = -mg. Note that the Maximum Height will Occur at a Time, tm, when the Velocity Goes to Zero, v(tm) = 0. a) Simply Solve the Equation of Motion for the Velocity as a Function of Time, and Determine the Time it Takes the Projectile to Reach the Maximum Height, tm, As a Function of the Given Parameters. b) Include a Force of Air Resistance that is Proportional to the Velocity of the Form Fr = -kmv. Again, Solve the Equation of Motion for the Velocity as a Function of Time, by Direct Integration or by Choosing an Appropriate Ansatz Form for the Solution. Again, Determine an Expression for the Time it Takes the Projectile to Reach the Maximum Height, tm, As a Function of the Given Parameters. c) Finally, Compare the Two Results by First Checking that they Agree for a Small Value of the Coefficient of Air Resistance, k→0, and, by a Taylor Series Expansion(Power Series) in the Small Parameter, k, Determine in this Limit if the Time it Takes the Projectile to Reach the Maximum Height is Shorter or Longer with the Air Resistance Included, and Give a Small Parameter Relationship For the Approximate Time Difference, Δt, Where Δt = tm|k-0 – tm|k = 0, Between the Two Flights to the Maximum Heights, which includes only the lowest power of k.
Recall that Newton’s Equation of Motion for a Body of Mass, m, is F = m(dv/dt). Assume the Initial Velocity of the Projectile is v0, and the Gravitational Acceleration Constant is g, where the Gravitational Force is Fg = -mg. Note that the Maximum Height will Occur at a Time, tm, when the Velocity Goes to Zero, v(tm) = 0.
a) Simply Solve the Equation of Motion for the Velocity as a Function of Time, and Determine the Time it Takes the Projectile to Reach the Maximum Height, tm, As a Function of the Given Parameters.
b) Include a Force of Air Resistance that is Proportional to the Velocity of the Form Fr = -kmv. Again, Solve the Equation of Motion for the Velocity as a Function of Time, by Direct Integration or by Choosing an Appropriate Ansatz Form for the Solution. Again, Determine an Expression for the Time it Takes the Projectile to Reach the Maximum Height, tm, As a Function of the Given Parameters.
c) Finally, Compare the Two Results by First Checking that they Agree for a Small Value of the Coefficient of Air Resistance, k→0, and, by a Taylor Series Expansion(Power Series) in the Small Parameter, k, Determine in this Limit if the Time it Takes the Projectile to Reach the Maximum Height is Shorter or Longer with the Air Resistance Included, and Give a Small Parameter Relationship For the Approximate Time Difference, Δt, Where Δt = tm|k-0 – tm|k = 0, Between the Two Flights to the Maximum Heights, which includes only the lowest power of k.
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