Answer:
The hammer must be moving at a speed of approximately 4.082 meters per second.
Explanation:
According to the statement and based on Principle of Energy Conservation, change in gravitational potential energy experimented by the metal piece ([tex]U_{g,o}[/tex]), in joules, must be equal to 18.8 percent of the translational kinetic energy of the hammer ([tex]K_{h}[/tex]), in joules.
[tex]U_{g, o} = 0.188\cdot K_{h}[/tex] (1)
By definitions of gravitational potential and translational kinetic energies, we expand (1):
[tex]m_{o}\cdot g\cdot h = 0.188\cdot \left(\frac{1}{2}\cdot m_{h}\cdot v^{2}\right)[/tex] (2)
Where:
[tex]m_{o}[/tex] - Mass of the metal piece, in kilograms.
[tex]g[/tex] - Gravitational acceleration, in meters per square second.
[tex]h[/tex] - Distance travelled by the metal piece, in meters.
[tex]m_{h}[/tex] - Mass of the hammer, in kilograms.
[tex]v[/tex] - Initial speed of the hammer, in meters per second.
If we know that [tex]m_{o} = 0.408\,kg[/tex], [tex]g = 9.807\,\frac{m}{s^{2}}[/tex], [tex]h = 4.23\,m[/tex] and [tex]m_{h} = 10.8\,kg[/tex], then the initial speed of the hammer is:
[tex]m_{o}\cdot g\cdot h = 0.188\cdot \left(\frac{1}{2}\cdot m_{h}\cdot v^{2}\right)[/tex]
[tex]10.638\cdot m_{o}\cdot g \cdot h = m_{h}\cdot v^{2}[/tex]
[tex]v = 3.261\cdot \sqrt{\frac {m_{o}\cdot g \cdot h}{m_{h}}}[/tex]
[tex]v = 3.261\cdot \sqrt{\frac{(0.408\,kg)\cdot \left(9.807\,\frac{m}{s^{2}} \right)\cdot (4.23\,m)}{10.8\,kg} }[/tex]
[tex]v \approx 4.082\,\frac{m}{s}[/tex]
The hammer must be moving at a speed of approximately 4.082 meters per second.
