Question

At a carnival, you can try to ring a bell by striking a target with a 8.13-kg hammer. In response, a 0.358-kg metal piece is sent upward toward the bell, which is 4.01 m above. Suppose that 17.0 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?

Answer 1

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Answer 2

To calculate the speed that the hammer needs to be moving to just barely ring the bell, we can use the conservation of energy principle. At the instant the hammer strikes the target, all of its initial kinetic energy will be transferred to the metal piece and the bell. We can write:

Kinetic energy of hammer before strike = Kinetic energy of metal piece and bell after strike

The kinetic energy of the hammer is given by:

K_hammer = (1/2) * m_hammer * v^2

where m_hammer is the mass of the hammer and v is its velocity.

The kinetic energy of the metal piece after the strike is given by:

K_metal = (1/2) * m_metal * v_metal^2

where m_metal is the mass of the metal piece and v_metal is its velocity.

The work done in lifting the metal piece to the height of the bell is given by:

W = m_metal * g * h

where g is the acceleration due to gravity (9.81 m/s^2) and h is the height of the bell (4.01 m).

We are told that 17% of the hammer's kinetic energy is used to do this work, so we can write:

W = 0.17 * K_hammer

Putting all of these equations together, we can solve for v:

(1/2) * m_hammer * v^2 = (1/2) * m_metal * v_metal^2 + 0.17 * (1/2) * m_hammer * v^2 + m_metal * g * h

Simplifying and solving for v, we get:

v = sqrt((2 * m_metal * g * h) / (0.83 * m_hammer))

Substituting the given values, we get:

v = sqrt((2 * 0.358 kg * 9.81 m/s^2 * 4.01 m) / (0.83 * 8.13 kg)) ≈ 8.29 m/s

Therefore, the hammer must be moving at a speed of about 8.29 m/s to just barely ring the bell.