Air Resistance & Horizontal Distance

[BravestIntent]

Air Resistance & Horizontal Distance

Warning: Calculus Required​

This was a discussion question in my class's Young & Freedman University Physics book. Page 150, Chapter 5, #29.

Q5-29 When a batted baseball (kicked football, whatever floats your boat, assume the initial vertical position was 0) moves with air drag, does it travel a greater horizontal distance while climbing to its maximum height or while descending from its maximum height back to the ground? Or is the horizontal distance traveled the same for both? Explain in terms of forces acting on the ball.

This is what I gave the class:
f=Dv^2 (or f=Dv for low velocity)

Does the angle at t=0 mater?

In general terms, it would spend more time in the air after peak height and before, yet it would not be moving as fast horizontally by that time.

Note that, simply put, it would take longer for the ball to fall from its peak height than to reach it (t1 < t2)...
The horizontal velocity before the peak height is achieved would be greater than afterwards (Vx1>Vx2) and this would decrease according to the formula... as is the horizontal force of air drag or wind resistance... and finally the absolute acceleration due to wind resistance in the horizontal direction.

Any contribution would be greatly appreciated.

 

[SGT]

I've ran a simulation with v_0 = 50 m/s, \theta_0 = 45^o and D = 0.02. The attached figure shows the result. The ball travels a greater horizontal distance during its way up than in the way down. It was expected, since the velocity is greater at the beginning of the trajectory than at the end.