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What will hit the ground first?

  1. Jun 21, 2004 #1
    Ok, classic old question. Two tennis balls dropped from a very high height. One normal, one filled with lead, for example.

    So what will hit first? I am certain that they will hit together (theoretically) by looking at the equation v = u + at so no mention of weight or mass. However equally there is no mention of air resistance.

    Now I know these are the same shape however when arguing my point someone said that the heavier ball would cut through the drag better. I can't really see this being true however I would like this confirmed.... So, basically my question is, in the real world what would happen?
    Last edited: Jun 21, 2004
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  3. Jun 21, 2004 #2


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    I'm not exactly sure, but I think you could consider something like a buoyant force (air is a fluid). Since this force doesn't depend on mass, only on volume, both balls would displace the same volume and thus be subject to the same buoyant force. However, a ball of less mass under a greater resistive (buoyant) force will accelerate towards the Earth at a lesser rate. Also, I don't know what other sorts of resistive or turbulent forces might come into effect, but they may be worth considering.
  4. Jun 21, 2004 #3
    Well the way I was thinking about it was that if I had a car in a wind tunnel and tested it with the engine in and then with the engine out, the results will be the same. Also, if the extra weight helped then surely the skydivers (the ones that dive for speed) would strap weights to their jump suits?
    Last edited: Jun 21, 2004
  5. Jun 21, 2004 #4
    The force air resistance should be the same for both. However, you'll note that the force of gravity is larger for the heavier ball than the lighter one. Thus, air resistance is comparatively smaller for the heavier one.

    Alternatively, you could look at it from a terminal velocity standpoint. Drag must negative gravity at terminal velocity. Gravity depends on m, drag depends on v (all other things being equal), so larger m yields a larger v.

  6. Jun 22, 2004 #5
    So the heavier ball has a greater terminal velocity, so if the balls fell long enough to reach terminal velocity then the heavier ball would hit the ground first?
  7. Jun 22, 2004 #6
    The heavier ball would land first on Earth because of air resistance. Although, if you dropped them in a vacuum they would fall at an equal rate and hit the ground at the same time.

  8. Jun 22, 2004 #7
    The heavier ball would hit the ground first. By Newton's second law the differential equation governing the motion of a ball of mass m in a uniform g field will be (aproximately)


    the second term on the left represents the air resistence. THe constant b is called aerodynamic coefficient or something and is determined experimentaly. Hence if the shape of the balls are equal their bs will be the same and the air resistance will be larger for the lighter ball, because the term proportional to the velocity is divided by m and hence larger when m is smaller.
  9. Jun 22, 2004 #8


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    If you leave the cars in neutral and allow the wind to push them backward, the one without an engine will start rolling first. Using this analogy, you can see wind resistance on your tennis balls as an acceleration. Since both objects have the same aerodynamic characteristics, and one has more mass, the one with more mass will gain less acceleration from the "wind".

    This requires changing your frame of reference, but from the balls' point of view, the ground is rushing toward them. This big wall of dirt rushing toward them is preceded by a gust of wind. Both balls are blown away from the approaching ground by this wind, but the lighter one is blown away faster. Therefore, the ground strikes the heavier one first and catches up with the lighter one later.
  10. Jun 22, 2004 #9


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    Don't mean to nit-pick here. Actually, the force of air resistance (drag) is the same for both balls. So the deceleration term from the air resistance is greater for the lighter ball.
  11. Jun 22, 2004 #10
    If you think of it as the ball colliding with air particles, you will see that the ball with greater mass will retain more momentum than the lighter ball will.
  12. Jun 24, 2004 #11
    As I see it, you just said the same thing I did but more elegantly and with less words!
  13. Jun 24, 2004 #12
    If the balls where in a vacuum they would both hit at the same time. In air the heavy ball would hit first because it has a greater terminal velocity.
  14. Jun 24, 2004 #13
    Imagine if you had 3 one pound metal balls, and when you droped them they all hit at the same time. Now take 2 of them and weld them together, and drop the 1 one pound and the now 1 two pound ball... What would happen? Are you saying the one little weld made the 2 balls, welded together, drop faster? That is kind of like saying if you were skydiving and you grab another skydivers hand, your weight would double so now you would instantly fall faster?
    Last edited: Jun 24, 2004
  15. Jun 24, 2004 #14


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    The fact the two hit at the same time is by no means dependent on the condition that there is air resistance. You would observe the same occurrence within a vacuum. Fundamentally, it has to do with the equivalence of gravitational and inertial mass. To see this; use F=ma, with Fg as the only acting force. Physically the m's on either side have different meanings (gravitational vs inertial mass) yet they are numerically equivalent, and thus cancel, leaving g=a, for any mass.
  16. Jun 24, 2004 #15


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    In welding them together, are you assuming they still have twice the volume and surface area? That's not the same as talking about a one pound ball and a two pound ball OF THE SAME VOLUME AND SURFACE AREA which is what everyone else is talking about.
  17. Jun 24, 2004 #16
    Are you sure two cars rolling because of wind is analogous to two tennis balls falling? Isn't the force you're trying to overcome to make the cars roll static friction? And isn't static friction determined by the (static coefficient)*Normal where Normal equals MASS times gravity? So by this argument the lighter car has a smaller frictional force to overcome before rolling, right?
  18. Jun 28, 2004 #17
    Anyone think of having one ball made of lead one one made of bismuth? At the same weight, in a vacuum, which one would hit the ground first? ;)

    (hint: bismuth repells magnetic fields)
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