Conservation of energy of a bow and arrow

[Sam Jelly]

Homework Statement

Imagine a drawn bow with an arrow, storing elastic potential energy. When we release the string, this elastic potential energy converts to kinetic energy, launching the arrow with a certain velocity. However, the bow itself also recoil due to the equal and opposite reaction force. Wouldn't this recoil give the bow some kinetic energy as well? If both the arrow and the bow are moving, doesn't that violate the law of conservation of energy? Shouldn't all the initial elastic potential energy be solely converted to the arrow's kinetic energy?

Relevant Equations

Total Energy = Constant

I thought the initial elastic potential energy would be converted to the arrow's kinetic energy but it also appears that the bow has some velocity.

 

[Orodruin]

Typically there is an archer holding the bow so that it does not recoil. If it does recoil then yes, the energy will be split between bow and arrow.

 

[berkeman]

How much movement of the bow do you see in her shot?



Can you comment about other lossy energy mechanisms in archery shooting? Can you comment about the differences in those losses for a compound bow versus a recurve bow?

 

[kuruman]

When we release the string, this elastic potential energy converts to kinetic energy, launching the arrow with a certain velocity. However, the bow itself also recoil due to the equal and opposite reaction force. Wouldn't this recoil give the bow some kinetic energy as well? If both the arrow and the bow are moving, doesn't that violate the law of conservation of energy? Shouldn't all the initial elastic potential energy be solely converted to the arrow's kinetic energy?

Let's change the situation a bit and imagine the archer standing on frictionless ice. He shoots the arrow standing up at rest with respect to the ice. The arrow moves forward and the bow + archer move backward to conserve linear momentum. The initial potential energy stored in the stretched bow is converted into kinetic energy of the arrow, bow and archer. It is not necessarily true that all the elastic potential energy must be converted to kinetic energy of the arrow.

However, when the archer stands on firm ground wearing cleated boots and there is no slipping, the bow + archer + Earth take up the recoil which means essentially no recoil . In this case, the arrow will travel farther than the archer-on-ice case because now it is true that all the elastic potential energy is converted into the arrow's kinetic energy.

The Mössbauer effect is an analog of this at the nuclear level. Nuclear transitions from an excited state to the ground state emit gamma rays (essentially) without recoil in which case the emitted gamma ray has essentially the same energy as the difference between the excited and ground state. That in itself may be just a curiosity until one realizes that these extremely monochromatic gamma rays can be used for nuclear gamma ray spectroscopy to probe the environment of nuclei in all sorts of interesting materials