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?
sqrt{2•0.411•9.8•5.22/0.262•8.46}
= 4.37 m/s
its still coming up wrong ?
To find the velocity at which the hammer must be moving to barely ring the bell, we can first calculate the kinetic energy of the hammer, and then use the fact that 34.4% of this energy is transferred to the metal piece.
Let's break down the steps to find the velocity of the hammer:
Step 1: Calculate the hammer's kinetic energy:
The kinetic energy of an object can be calculated using the equation:
Kinetic Energy (KE) = (1/2) * mass * velocity^2
In this case, the mass of the hammer (m) is given as 8.46 kg. Let's assume the velocity of the hammer is v, and the kinetic energy of the hammer is KE_hammer.
So, KE_hammer = (1/2) * 8.46 kg * v^2
Step 2: Calculate the work done on the metal piece:
The work done on the metal piece is equal to the kinetic energy transferred to it. In this case, 34.4% of the kinetic energy of the hammer is used to lift the metal piece. Let's assume the mass of the metal piece is m_piece, the acceleration due to gravity is g, and the work done on the metal piece is W.
Since the metal piece is lifted vertically upwards, the work done can be expressed as:
W = m_piece * g * height
In this case, the mass of the metal piece (m_piece) is given as 0.411 kg, the acceleration due to gravity (g) is approximately 9.8 m/s^2, and the height (h) is given as 5.22 m.
Step 3: Calculate the velocity of the hammer:
We know that 34.4% of the kinetic energy of the hammer is equal to the work done on the metal piece. Mathematically, we can express this as:
0.344 * KE_hammer = W
Since we have calculated the expression for KE_hammer in terms of the hammer's velocity (v) in Step 1, we can substitute it into the above equation to solve for v.
0.344 * [(1/2) * 8.46 kg * v^2] = 0.411 kg * 9.8 m/s^2 * 5.22 m
Simplifying the equation:
0.172 * 8.46 kg * v^2 = 2.075 kg * m^2/s^2
Dividing both sides of the equation by (0.172 * 8.46 kg), we get:
v^2 = (2.075 kg * m^2/s^2) / (0.172 * 8.46 kg)
Simplifying further:
v^2 ≈ 1.12633 m^2/s^2
Finally, taking the square root of both sides of the equation, we find:
v ≈ 1.061 m/s
Therefore, the hammer must be moving at approximately 1.061 m/s when it strikes the target to barely ring the bell.