Problem 1C-09***: Two-dimensional free throw problem with air resistance

In Problem 1A-09 we considered a vertically thrown ball with air resistance. The equation of motion for the vertical position of the ball, y(t), is given as

where g and ε are given positive constants. In this problem the position of the ball is described by two time-dependent functions, the vertical position function y(t) and the horizontal position function x(t). Eq.(1) still applies to y(t). A new equation of motion is needed for x(t), and it is
Suppose that the ball is thrown with an initial speed of u from a height of h0, and the ball is thrown with an angle θ with regard to a horizontal line.

(a) What is the maximum height that the ball will reach?

(b) Let the horizontal distance that the ball will travel be denoted as L. Calculate L in terms of given constants, g, ε, u, h0 and θ under the assumption of εT ‹‹ 1, where T is the flight time of the ball.

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(a) Let the velocity components of the ball be denoted as vy(t) = dy/dt and vx(t) = dx/dt respectively. Eqs.(1) and (2) then can be written as

As discussed in Problems 1A-08 and 1A-09, the general solution of Eqs.(3) are

Integrating both sides of two equations of Eqs.(4), we get

Initial conditions vy(0) = u·sinθ and y(0) = h0 lead to A = g/ε+u·sinθ and C = h0+(A/ε). Initial conditions vx(0) = u·cosθ and
x(0) = 0 lead to B = u·cosθ and D = (u/ε)·cosθ. Eqs.(5) and (4) become

Let the time that the ball reaches its maximum height be denoted as τ. To calculate τ, we use the conditin vy(τ) = 0. So we get
The maximum height H that the ball reaches is then calculated as

We note that this result is the same as that of Problem 1A-09.

(b) To find the flight time of the ball T, we need to solve the equation y(T) = 0. The equation is

The solution of this equation can not be written in a known simple function. If the numerical values of all given constants are available, a computer can be used to calculate the value of T. For the general case, nothing more can be said about T. However, if the condition εT ‹‹ 1 is imposed, then the exponential function can be approximated as
The equation of concern then becomes
This can be simplified to

From Eq.(7) we know

Since L = x(t), we have from the above equation and Eq.(6) that
where T is given in Eq.(8).

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