At a carnival, you can try to ring a bell by striking a target with a 8.46-kg hammer. In response, a 0.411-kg metal piece is sent upward toward the bell, which is 5.22 m above. Suppose that 34.4 percent of the hammer's kinetic energy is used to do the work of sending the metal piece upward. How fast must the hammer be moving when it strikes the target so that the bell just barely rings

0.344KE=PE
0.344m₁v²/2 = m₂gh

sqrt{2•0.411•9.8•5.22/0.262•8.46}

= 4.37 m/s

its still coming up wrong ?

(2•0.411•9.8•5.22/0.344•8.46)

work done on metal = m g h = .411*9.81*5.22

= 21.05 Joules

21.05 = .344 ke of hammer
so
ke of hammer = 61.2 Joulesd

(1/2) 8.46 v^2 = 61.2

v^2 = 14.5
v = 3.80 m/s

To solve this problem, we need to apply the principle of conservation of energy. The kinetic energy of the hammer just before it strikes the target will be used to do work on the metal piece and raise it to a certain height.

Let's go step-by-step to find the correct solution:

1. We are given:
- Mass of the hammer (m₁) = 8.46 kg
- Mass of the metal piece (m₂) = 0.411 kg
- Height of the bell (h) = 5.22 m
- Percentage of kinetic energy used (0.344)

2. The total initial kinetic energy (KE) of the hammer is given by:
KE = (1/2)mv², where v is the speed of the hammer just before striking the target.

3. According to the conservation of energy principle, the kinetic energy used to raise the metal piece is equal to the potential energy gained by the metal piece at height h:
0.344KE = m₂gh
Rearranging the equation, we get:
KE = (m₂gh) / 0.344

4. Substituting the given values, we get:
KE = (0.411 kg)(9.8 m/s²)(5.22 m) / 0.344
KE = 25.04 J

5. Rearranging the equation for KE, we have:
KE = (1/2)m₁v²
Solving for v², we get:
v² = (2KE) / m₁

6. Substituting the values, we get:
v² = (2)(25.04 J) / 8.46 kg
v² = 5.92 m²/s²

7. Taking the square root of both sides, we find:
v ≈ 2.43 m/s

Therefore, the hammer must be moving at approximately 2.43 m/s when it strikes the target for the bell to just barely ring.