Double ball attempt

Visualization of the ball pyramid

The double ball experiment, also known as the ball pyramid or super jump , is a physical experiment that illustrates conservation of momentum . Two or more balls are placed on top of each other and dropped from a certain height, the lower ball being heavier than the upper one. It can be observed that after the impact on the ground, the top and lightest ball bounces well above its starting height. After hitting the ground, the lower, heavier ball transfers its momentum to the upper, lighter ball, which greatly increases its speed.

With the help of the physical toy “Astroblaster” from Stirling Colgate , this structure illustrates in a very simplified way how matter is distributed during the core collapse of a supernova .

Play media file

Video of the double ball attempt

Setup and observation

If a single "idealized" ball, which is elastic , frictionless and perfectly round, moves in free fall onto an idealized floor that does not absorb any energy, it will again reach its original height. Real balls also emit energy through friction and irreversible deformation , so that there is a difference between the initial and final height: the ball does not reach its initial height after ricocheting. Based on this everyday experience, it is surprising that when a ball falls, it can reach a greater height than its starting height.

In the double ball attempt, two balls are placed vertically on top of each other, with the top ball being lighter than the ball below. For stability, the balls can be connected in the middle with a string or a rod so that the center of gravity of the balls and their contact point are on one line. The balls must not be fixed, they must still be able to move separately from each other. If these balls are dropped, one can observe that the lower, heavier ball bounces up only a little. At the same time, the upper, lighter ball bounces up far above its original height.

In the extended experiment, in which several balls are arranged on top of each other, one speaks of a ball pyramid.

Conservation of momentum

Similar to the shot-put pendulum , the simultaneous conservation of energy and momentum is crucial to understanding this experiment. The momentum of a moving object is the product of its mass and its speed (“mass times speed”). If the lower ball hits the ground, it is compressed and expands again. The direction of his impulse is reversed and he moves up again. When it hits the second ball, which is still falling, the heavier ball transfers its momentum to the lighter one. Thus, the heavier ball is slowed down and falls back to the ground relatively soon. Since the second ball is lighter (i.e. has a lower mass), its speed increases because the total momentum of the two balls is retained.

calculation

Ideal case - two balls

Assuming that there are two ideal balls with purely elastic impacts , the final speed and height can be calculated directly from the conservation of energy and momentum. This results in

{\ displaystyle v_ {2} '= {\ frac {m_ {2} \ cdot v_ {2} + m_ {1} \ cdot (2v_ {1} -v_ {2})} {m_ {1} + m_ { 2}}}}

With

${\ displaystyle v_ {1}}$ : Speed of the heavier ball before the kick
${\ displaystyle v_ {2}}$ : Speed of the lighter ball before the kick
${\ displaystyle v_ {2} '}$ : Speed of the lighter ball after the shot
${\ displaystyle m_ {1}}$ : Mass of the heavier ball
${\ displaystyle m_ {2}}$ : Mass of the lighter ball

Assuming that both balls now have the same speed with opposite directions shortly before the impact,

{\ displaystyle v_ {1} = v = -v_ {2},}

the following results for the speed of the lighter ball after the shot:

{\ displaystyle v_ {2} '= v \ cdot {\ frac {3m_ {1} -m_ {2}} {m_ {1} + m_ {2}}} \ qquad}

The maximum height is reached when the entire kinetic energy of the ball has been converted into potential energy . The following applies to the ratio of the jumping height for the lighter ball to its starting height : ${\ displaystyle h_ {2}}$ ${\ displaystyle h_ {0}}$

{\ displaystyle {\ frac {h_ {2}} {h_ {0}}} = \ left ({\ frac {3m_ {1} -m_ {2}} {m_ {1} + m_ {2}}} \ right) ^ {2}}

If the same applies to the masses , the lighter ball will reach three times its initial speed and thus nine times its initial height. ${\ displaystyle m_ {1} \ gg m_ {2}}$

Real case - two balls

Energy losses due to the deformation of the heavier ball when it hits the ground are taken into account with the impact number . The lower ball then has a speed ${\ displaystyle k}$

{\ displaystyle v_ {1} = k \ cdot v,}

after bouncing off the ground. For the speed of the lighter ball after the impact with the heavier one results:

{\ displaystyle v_ {2} '= v \ cdot {\ frac {m_ {1} (k ^ {2} + 2k) -m_ {2}} {m_ {1} + m_ {2}}}}

In the real case, energy losses occur, so is. In the ideal case , what again leads to the formula for given above applies . ${\ displaystyle k <1}$ ${\ displaystyle k = 1}$ ${\ displaystyle v_ {2} '}$

Real three ball case

If the calculation is continued on three balls, the impact of ball 2 and ball 3 must also be considered, with ball 2 being the middle ball and ball 3 being the lightest ball. Again the law of momentum and energy conservation applies to the collision. For the speed of the third ball after the hit with the second we get:

{\ displaystyle v_ {3} '= {\ frac {v} {1 + q_ {2}}} \ cdot \ left ({\ frac {(1 + k) (k ^ {2} + 2k-q_ {1 })} {1 + q_ {1}}} + k-q_ {2} \ right)}

The mass ratios and were used. ${\ displaystyle q_ {1} = {\ tfrac {m_ {2}} {m_ {1}}}}$ ${\ displaystyle q_ {2} = {\ tfrac {m_ {3}} {m_ {2}}}}$

If one considers the ideal case of a completely elastic collision with and , one obtains a speed and a final height of the third ball which corresponds to 49 times the initial height. ${\ displaystyle k = 1}$ ${\ displaystyle m_ {1} \ gg m_ {2} \ gg m_ {3}}$ ${\ displaystyle v_ {3} '= 7 \ cdot v}$

For a pyramid made up of other balls, the calculation can be carried out in the same way by considering the impacts of two neighboring balls.

Illustration of processes in a supernova with the "astroblaster"

In the case of collapsing stars, there are mechanisms that distribute matter at a high rate of expansion in supernovae . Here, the outer shell of the star, which consists of light elements, is accelerated outwards to a multiple of its initial speed when it hits the collapsing inner shell, such as the light ball in the ball pyramid. The game "Astroblaster" by Stirling Colgate is based on this analogy . The toy consists of four balls of different weights that are lined up on a stick. The heaviest ball is firmly attached to a stick, while the two middle balls can move on the stick. Since the rod has a larger diameter at the end, the two middle balls are prevented from sliding off the rod. Only the top and lightest ball can be removed from the stick. Part of this stick protrudes above the balls and helps keep the ball pyramid perpendicular to the ground. If this is now released, the ball pyramid falls to the ground and the lightest ball rises to over ten times its starting height.

literature

Norbert Treitz: Bridge to Physics. Harri Germany, Thun / Frankfurt am Main 1997, ISBN 3-8171-1518-0 , pp. 119-120.
Norbert Treitz: Easy game with the focus. In: Spectrum of Science. August 2004, pp. 101-104.
Norbert Treitz: One push gives another. In: Spectrum of Science. March 2005, pp. 114-117.
Jearl Walker : The Flying Circus of Physics. Oldenbourg Verlag, Munich 2008, ISBN 978-3-486-58067-9 , pp. 22-23.

Web links

Video demonstration and explanation of the ball pyramid with two and three balls of Dianna Cowern on YouTube (English)
Simulation of the experiment with two balls by Leifi Physik
Derivation of the calculations in the real case for two or more balls from the Munich University of Applied Sciences

Individual evidence

↑ ^a ^b ^c ^d Jörg Hüfner and Rudolf Löhken: Lecture 5: “Ball Games”. In: Lecture: “Physics is everywhere” 2007/08. Retrieved July 12, 2017 .
↑ ^a ^b ^c Saint Mary's University in Halifax Physics Demos: Double Ball Drop. Retrieved August 1, 2017 .
↑ ^a ^b Jearl Walker: The flying circus of physics . 9th edition. Oldenbourg, Munich 2008, ISBN 978-3-486-58067-9 , pp. 22-23 .
^ ^A ^b University of Virginia Physics Show: Double Ball Bounce. Retrieved August 1, 2017 .
↑ ^a ^b Ball pyramid: elastic collisions between unequal masses. Physics lecture collection, Ulm University, accessed on August 1, 2017 .
↑ Central straight elastic collision. Learning aid, accessed on July 12, 2017 .
↑ Conservation of momentum and shocks - double ball. LeifiPhysik, accessed March 27, 2020 .
↑ ^a ^b ^c ^d ^e Reflection of two or more balls on the ground (super jump). Munich University of Applied Sciences, accessed on July 5, 2017 .
↑ Norbert Treitz: Easy game with the focus. In: Spectrum of Science. August 2004, pp. 101-104.
↑ Wolfgang Demtröder: Experimentalphysik 4 . 5th edition. Springer Spectrum, 2017, ISBN 978-3-662-52883-9 , pp. 341 .
^ Seismic Accelerator. Educational Innovations Inc., accessed July 12, 2017 .
↑ Marián Kires: Astro Blaster - a fascinating game of multi-ball collisions . In: Physics Education . tape 44 , no. 2 , March 2009, p. 159 , doi : 10.1088 / 0031-9120 / 44/2/007 .

This version was added to the list of articles worth reading on September 28, 2017 .

[:0-1] Jörg Hüfner and Rudolf Löhken: Lecture 5: “Ball Games”. In: Lecture: “Physics is everywhere” 2007/08. Retrieved July 12, 2017 .

[:2-2] Saint Mary's University in Halifax Physics Demos: Double Ball Drop. Retrieved August 1, 2017 .

[:4-3] Jearl Walker: The flying circus of physics . 9th edition. Oldenbourg, Munich 2008, ISBN 978-3-486-58067-9 , pp. 22-23 .

[:3-4] A ^b University of Virginia Physics Show: Double Ball Bounce. Retrieved August 1, 2017 .

[:5-5] Ball pyramid: elastic collisions between unequal masses. Physics lecture collection, Ulm University, accessed on August 1, 2017 .

[6] Central straight elastic collision. Learning aid, accessed on July 12, 2017 .

[7] Conservation of momentum and shocks - double ball. LeifiPhysik, accessed March 27, 2020 .

[:1-8] Reflection of two or more balls on the ground (super jump). Munich University of Applied Sciences, accessed on July 5, 2017 .

[9] Norbert Treitz: Easy game with the focus. In: Spectrum of Science. August 2004, pp. 101-104.

[10] Wolfgang Demtröder: Experimentalphysik 4 . 5th edition. Springer Spectrum, 2017, ISBN 978-3-662-52883-9 , pp. 341 .

[11] Seismic Accelerator. Educational Innovations Inc., accessed July 12, 2017 .

[12] Marián Kires: Astro Blaster - a fascinating game of multi-ball collisions . In: Physics Education . tape 44 , no. 2 , March 2009, p. 159 , doi : 10.1088 / 0031-9120 / 44/2/007 .