Calculating Combined Mass of Archer & Bow w/ Conservation of Momentum

In summary: With your initial and final you describe a fully inelastic collision of an arrow and an archer plus bow. Rather fatal for the archer ...Is it strange you get a negative mass ? The recoil from shooting the arrow makes the archer go backwards (that's why it's called recoil) wrt the arrow ! Yet you have the same sign for both speeds !In summary, an archer stands on a frictionless surface and shoots an arrow forward with an initial velocity of 92m/s. The combined mass of the archer and the bow is -66.05128205kg.
  • #1
Balsam
226
8

Homework Statement


After shooting a 28g arrow with an initial velocity of 92m/s[forward], an archer standing on a frictionless surface travels in the opposite direction at a speed of 0.039m/s. Calculate the combined mass of the archer and the bow.

Given: **Subscript of 1 indicates values for the arrow and subscript of 2 indicates values for the archer

m1=0.028kg
vi1=92m/s[forward]
vf1=0m/s

vi1=0.039m/s[backward]
vf2=0m/s

Homework Equations


m1vi1+m2vi2=m1vf1+m2vf2

The Attempt at a Solution


The only way this problem works is if you treat the archer and the bow as one body/one object and using the above formula and values, solve for m2:

(0.029)(92)+m2(0.039)=0
2.576=-m2(0.039)
-66.05128205kg=m2

I got the correct answer aside from the negative sign in front of my answer. Why is my answer negative? Does that mean I did something wrong?

Also, why do you treat the archer and the bow as one collective object? I thought the law of conservation of momentum, which gives us the equation I used, applied to situations when two objects collide in an isolated system. If so, then what collision occurs in this problem; how are both the bow and the archer colliding with the arrow?
 
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  • #2
With your initial and final you describe a fully inelastic collision of an arrow and an archer plus bow. Rather fatal for the archer ...

Is it strange you get a negative mass ? The recoil from shooting the arrow makes the archer go backwards (that's why it's called recoil) wrt the arrow ! Yet you have the same sign for both speeds !
 
  • #3
BvU said:
With your initial and final you describe a fully inelastic collision of an arrow and an archer plus bow. Rather fatal for the archer ...

Is it strange you get a negative mass ? The recoil from shooting the arrow makes the archer go backwards (that's why it's called recoil) wrt the arrow ! Yet you have the same sign for both speeds !

I thought that it was okay not to plug in negative signs indicating direction for vector values when doing calculations to find a scalar value, like mass.
 
  • #4
Well, apparently you thought wrong: a negative mass is unphysical !
 
  • #5
BvU said:
Well, apparently you thought wrong: a negative mass is unphysical !
Good to know
 
  • #6
BvU said:
With your initial and final you describe a fully inelastic collision of an arrow and an archer plus bow. Rather fatal for the archer ...

Is it strange you get a negative mass ? The recoil from shooting the arrow makes the archer go backwards (that's why it's called recoil) wrt the arrow ! Yet you have the same sign for both speeds !
But why are the bow and archer one mass?
 
  • #7
Balsam said:
But why are the bow and archer one mass?
Because the archer is assumed to hold the bow rigidly. In a different model, the archer could let go of bow and arrow at the same instant. The arrow would travel a smallish distance one way, and the bow a smaller distance the other.
Note that with a firearm that second model is closer to what happens. The impulse is so great that it is not possible (and maybe not a good idea to try) to hold the weapon rigidly.
 
  • #8
haruspex said:
Because the archer is assumed to hold the bow rigidly. In a different model, the archer could let go of bow and arrow at the same instant. The arrow would travel a smallish distance one way, and the bow a smaller distance the other.
Note that with a firearm that second model is closer to what happens. The impulse is so great that it is not possible (and maybe not a good idea to try) to hold the weapon rigidly.
What is the collision that happens, the bow and the arrow are not colliding, they were already in contact.
 
  • #9
Balsam said:
What is the collision that happens, the bow and the arrow are not colliding, they were already in contact.
Conservation of momentum is not limited to collisions.
 
  • #10
Balsam said:
What is the collision that happens
Physically the time reversal of a collision happens: first they are together (sum of momenta = 0) and then they separate -- without any external forces acting. So the center of mass does not accelerate and sum of momenta remains zero.

Newton 3: action = - reaction makes that afterwards the individual momenta are equal and opposite.
 

Related to Calculating Combined Mass of Archer & Bow w/ Conservation of Momentum

1. How do you calculate the combined mass of an archer and a bow?

The combined mass of an archer and a bow can be calculated by adding the mass of the archer to the mass of the bow. The mass of the archer can be measured using a scale, while the mass of the bow can be determined by using the formula density = mass/volume and measuring the volume of the bow.

2. What is conservation of momentum and how does it apply to an archer and a bow?

Conservation of momentum is a fundamental law of physics that states that the total momentum of a closed system remains constant. In the case of an archer and a bow, the momentum of the arrow is equal to the combined momentum of the archer and the bow before the arrow is released.

3. How does the velocity of the arrow relate to the combined mass of the archer and the bow?

The velocity of the arrow is directly proportional to the combined mass of the archer and the bow. This means that as the combined mass increases, the velocity of the arrow also increases, resulting in a more powerful shot.

4. Can the combined mass of an archer and a bow affect the accuracy of the shot?

Yes, the combined mass of an archer and a bow can affect the accuracy of the shot. A heavier combined mass can result in more recoil and make it difficult for the archer to maintain a steady aim. However, a heavier combined mass can also provide more stability and reduce the effects of external factors such as wind.

5. Are there any other factors that can affect the combined mass of an archer and a bow?

Aside from the mass of the archer and the bow, other factors that can affect the combined mass include the type of arrow being used, the accessories attached to the bow (such as sights or stabilizers), and the technique and strength of the archer. These factors should also be taken into consideration when calculating the combined mass for a more accurate result.

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