The pile driver struck the piling with a speed of 20 m/s. At what height did the pile driver hammer have to be lifted for it to fall freely and attain that speed before striking the piling?
Show steps please!!!
potential energy at start=Kenergy at end
mgh=1/2 m v^2
h= 1/2 v^2/g
To determine the height the pile driver hammer has to be lifted, we can use the principle of conservation of energy. The potential energy of the hammer when lifted to a certain height is converted into kinetic energy as it falls freely.
The potential energy (PE) of an object is given by the equation:
PE = mgh
where m is the mass of the hammer, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
The kinetic energy (KE) of an object is given by the equation:
KE = (1/2)mv²
where v is the speed of the hammer (20 m/s).
Since the potential energy is converted into kinetic energy, we can equate the two equations:
mgh = (1/2)mv²
Canceling the mass, m, on both sides of the equation, we get:
gh = (1/2)v²
Now, solve for h by dividing both sides of the equation by g:
h = (1/2)v² / g
Plugging in the given values of v = 20 m/s and g = 9.8 m/s², we can calculate the height:
h = (1/2)(20² m²/s²) / 9.8 m/s²
h = 200 m²/s² / 9.8 m/s²
h = 20.41 meters (rounded to two decimal places)
Therefore, the pile driver hammer needs to be lifted to a height of approximately 20.41 meters for it to fall freely and attain a speed of 20 m/s when striking the piling.
To find the height at which the pile driver hammer must be lifted for it to attain a speed of 20 m/s before striking the piling, we can use the principle of conservation of energy. Here are the steps:
Step 1: Determine the gravitational potential energy of the hammer at the initial height.
The gravitational potential energy (PE) is given by the equation: PE = m * g * h
where m is the mass of the hammer, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
Step 2: Determine the kinetic energy of the hammer when it strikes the piling.
The kinetic energy (KE) is given by the equation: KE = 0.5 * m * v^2
where m is the mass of the hammer and v is the velocity (speed) of the hammer, which is given as 20 m/s.
Step 3: Equate the initial potential energy to the final kinetic energy.
PE = KE
m * g * h = 0.5 * m * v^2
Step 4: Cancel out the mass (m) on both sides of the equation.
g * h = 0.5 * v^2
Step 5: Solve for the height (h).
h = (0.5 * v^2) / g
Substituting the given values:
h = (0.5 * 20^2) / 9.8
Calculating:
h = (0.5 * 400) / 9.8
h = 200 / 9.8
h ≈ 20.41 meters
Therefore, the pile driver hammer must be lifted to a height of approximately 20.41 meters for it to fall freely and attain a speed of 20 m/s before striking the piling.