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Name: David
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Question:
My name is David, and I am currently stationed in Iraq. Myself and about ten other soldiers are having an argument that if you drop two different objects from the same height at the same time, they will or will not land at the same time. For example, let us say that a marble and a sixteen pound bowling ball are dropped at the same time from two thousand feet in the air. Which would land first, and if not at the same time, why?



Replies:
Hi,

In vacuum (that is, in the absence of any medium such as air) objects would fall at the same speed under gravity. But in a medium other than vacuum, such as in air or water, their speeds vary, impacted by the resistance they encounter in pushing through the medium. This resistance is related to the cross section area of the object. That explains why an open parachute drops down much slower than one that is not opened:. same weight but different cross-sections. You can experience the dependence of resistance on cross- sectional area by holding a hand out through the window of a moving vehicle.

When the palm is perpendicular to the direction of motion, it is pushed back harder then when it is parallel.

Ali Khounsary, Ph.D.
Advanced Photon Source
Argonne National Laboratory


David,

If the objects were in a vacuum, such as being on the Moon, they would both fall at exactly the same rate. The falling would be due to nothing but gravity. Because gravitational force is proportional to mass, the resulting acceleration is the same for all objects.

If the objects are in a material such as air, other things play a part. Air resistance depends on size, shape, and speed. Air resistance is NOT proportional to mass. A ping-pong ball will fall much more slowly than either a marble or a bowling ball.

A good demonstration requires a hard-cover book and a flat piece of paper smaller than the book. Do all drops with the book/paper horizontal, with the large surface (e.g. front cover) toward the ground. When dropped separately, the book reaches the ground much more quickly that the paper. When dropped with the paper flat against the bottom side of the book, they fall together. This makes sense, since we see the book drop faster than the paper. The third case requires a little more thought. Put the paper flat against the top side of the book. What do you expect to see? Since air resistance results from an object pushing air molecules out of the way, how much air does the paper actually have to push?

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College


Air resistance enters into the problem at once. The bowling ball has more mass (weight) per unit of surface area than the marble so it will punch, if you will, through the air, far more effectively than the marble. The relationship between weight and surface area is very important in deciding what fall faster.

A weird example would be: Which will fall faster, a feather or an exact copy of that feather made of lead? The lead feather, right? It has more weight per surface area.

Same thing goes for people. Stockier, heavier folks have higher terminal velocities when free falling than the tall, skinny ones.

Bob Avakian
Oklahoma State University - Okmulgee


The classic answer is that all objects fall at the same rate of acceleration (due to gravity), but this is only in a vacuum (e.g. no air). In a vacuum, a feather would fall at the same rate as a lead ball. With air, wind resistance makes a big difference. Obviously a feather falls more slowly than a lead ball when you have air. If you drop a tennis ball and basketball from shoulder height at the same time, you're not going to see very much of a difference. If you drop them both from an airplane, the differences become more noticeable.

In the case of a objects of different weight and size, it gets even more complicated. When an object falls, there are three forces to consider: the pull of gravity, the drag force, and the buoyant force. In air, the buoyant force is usually small enough to ignore. Gravity just depends on the mass of the object (although technically the force of gravity changes slight from location to location). The last force, drag, is the complicated one.

Drag force on a sphere depends on the size, mass, and speed at which the sphere is falling. When the object first starts to fall, or if it is very small, air can flow around it smoothly. At a certain point, if the ball is large, or if it is moving fast enough, air can't flow smoothly around it anymore; the air flow will become turbulent.

When the air is flowing smoothly (called 'laminar' flow), the drag force is proportional to the radius of the sphere and to its velocity. Double the radius, double the drag force. Double the velocity, double the drag force. This is known as Stokes Drag -- you can look it up on the internet, including a maddeningly complicated mathematical derivation of the drag force.

When you have a very large object moving very fast, the air flow becomes turbulent, and the drag changes. In this regime, called Quadratic Drag, drag force is proportional to the square of velocity and to the cross-sectional area of the object.

Worse, as the ball accelerates, depending on its size and weight, it may transition from one regime to another. So it's very hard to actually predict how fast/how long it will take when falling a long distance.

There is a number that describes the combination of size and speed, and predicts when the transition from smooth to turbulent flow will occur called the Reynolds Number. To really understand the science, I recommend you look up "stokes drag", "quadratic drag", and "Reynolds number" on the internet. You'll have many options, ranging from basic college physics classes to highly technical scientific publications.

Lastly, all of the above assume the air is constant density and is not moving. In a real situation, the air gets less dense with greater altitude (and less dense air means lower Reynolds number), so it also depends not only how far you drop, but where you drop from. If you have air currents, they will also affect things.

To answer your specific question about which ball would land first, as you might have already predicted, it depends on what balls you use and how far you let them fall. In the Stokes regime (e.g. low Reynolds number / small balls traveling slowly), a larger ball will fall faster than a smaller ball assuming the same material. That is because the gravity force increases with the cube of radius, but the drag force only proportionally with radius. A larger/faster moving ball would transition into the quadratic drag sooner than the smaller/slower ball. My intuition tells me the bowling ball would fall faster than the marble, but intuition is often wrong, so I would want to do the calculation before answering with certainty (actually, I want to do an experiment, but that might be unsafe...). Doing the math is not the easiest thing to do, but armed with this information is certainly possible (try it!).

This is a bit complicated, so in the interest of being brief, I've glossed over some details. If you want more explanation, feel free to reply with more questions.

Hope this helps,

Burr Zimmerman



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