A piledriver of mass 5kg falls through a height of 20m into a pile of mass 3kg and the pile is driven a distance of 10cm into the ground. Calculate the acceleration with which the pile is driven into the ground.
To calculate the acceleration with which the pile is driven into the ground, we can use the principle of conservation of energy.
First, let's calculate the total initial potential energy of the piledriver:
Potential energy = mass * gravity * height
Potential energy = 5kg * 9.8m/s^2 * 20m
Potential energy = 980J
Next, let's calculate the total final energy of the system (piledriver + pile) after the collision:
Final energy = kinetic energy of the piledriver + kinetic energy of the pile
The kinetic energy for each object can be calculated as:
Kinetic energy = 0.5 * mass * velocity^2
Since the piledriver falls through a height of 20m, it gains velocity equal to the free-fall velocity. We can calculate it as:
Velocity = sqrt(2 * gravity * height)
Velocity = sqrt(2 * 9.8m/s^2 * 20m)
Velocity ≈ 19.8 m/s
Using the kinetic energy equation, the kinetic energy of the piledriver is:
Kinetic energy of piledriver = 0.5 * 5kg * (19.8m/s)^2
Kinetic energy of piledriver ≈ 988 J
After the collision, the piledriver and the pile form a single system. The kinetic energy of this system is the sum of the kinetic energy of the piledriver and the pile. Since no external forces are acting on the system, this energy is conserved.
Thus, the final energy of the system is also 988 J.
Now, let's calculate the energy lost during the collision:
Energy lost = Initial potential energy - Final energy
Energy lost = 980J - 988J
Energy lost = -8J (negative because energy is lost)
The energy lost during the collision is converted into work done by the pile driving into the ground. This work can be calculated as:
Work = force * distance
We know the distance (10cm = 0.1m) and we can find the force using the energy lost:
Work = -8J = force * 0.1m
force = -8J / 0.1m
force = -80 N (negative because the force acts in the opposite direction)
Finally, we can use Newton's second law of motion, F = ma, to calculate the acceleration:
acceleration = force / mass of the pile
acceleration = -80N / 3kg
acceleration ≈ -26.7 m/s^2
Therefore, the acceleration with which the pile is driven into the ground is approximately -26.7 m/s^2.