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.

 

[quicknote]

I would like to clarify a few things before providing a response to this question. Firstly, the statement "Warning: Calculus Required" is not entirely accurate. While calculus may be necessary for a more detailed analysis, this question can still be answered using basic principles of physics.

Now, to answer the question at hand. When a batted baseball moves with air drag, the horizontal distance it travels will depend on several factors such as the initial velocity, launch angle, and air resistance. However, in general, the horizontal distance traveled will be greater while the ball is descending from its maximum height back to the ground.

To understand why, we need to look at the forces acting on the ball. When the ball is at its maximum height, the only force acting on it is gravity, which is pulling it towards the ground. As the ball starts to descend, the force of air resistance also comes into play, acting in the opposite direction of the ball's motion. This force increases as the ball gains speed, ultimately slowing it down.

On the other hand, when the ball is climbing to its maximum height, both gravity and air resistance are acting in the same direction, which is towards the ground. This means that the ball will experience a greater net force and will reach its peak height faster. As it starts to descend, the force of air resistance will slow it down, resulting in a shorter horizontal distance traveled.

In summary, the horizontal distance traveled by a batted ball with air resistance will be greater while descending from its maximum height back to the ground due to the opposing forces of gravity and air resistance. This can also be seen in the formula provided, where the horizontal force of air drag decreases as the ball slows down, resulting in a shorter horizontal distance traveled.