While chopping down his fathers cherry tree; George discovered that if he swing the axe with a speed of 25 m/s it would embed itself 2.3 cm into the tree before coming to a stop. If the axe head has a mass of 2.5 kg, how much force was the tree exerting on the axe head upon impact? How much did the ace exert back on the tree?
To determine the force exerted by the tree on the axe head upon impact, we can use the principle of work and energy.
The work done on the axe head by the tree is equal to the change in kinetic energy of the axe head. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy.
The initial kinetic energy of the axe head is given by:
KE_initial = (1/2) * mass * velocity^2
KE_initial = (1/2) * 2.5 kg * (25 m/s)^2
The final kinetic energy of the axe head is zero, as it comes to a stop. Therefore, the change in kinetic energy is equal to the initial kinetic energy:
Change in KE = KE_final - KE_initial
Change in KE = 0 - ((1/2) * 2.5 kg * (25 m/s)^2)
Since the work done on the axe head by the tree is equal to the change in kinetic energy, we have:
Work = Change in KE
Work = -((1/2) * 2.5 kg * (25 m/s)^2)
Now, we can calculate the work done by the tree. However, work is equal to force multiplied by distance (W = F * d), so we need to find the distance the axe head moved into the tree upon impact.
The displacement of the axe head into the tree is given as 2.3 cm, which is 0.023 meters.
Therefore, the work done by the tree can be expressed as:
-((1/2) * 2.5 kg * (25 m/s)^2) = F * 0.023 m
Rearranging the equation to solve for force (F), we have:
F = -((1/2) * 2.5 kg * (25 m/s)^2) / 0.023 m
Evaluating the expression, we can find the force exerted by the tree on the axe head upon impact.
To find the force exerted back on the tree, we can use Newton's third law of motion, which states that every action has an equal and opposite reaction. Therefore, the force exerted by the axe on the tree upon impact is equal in magnitude but opposite in direction to the force exerted by the tree on the axe head.
Hence, the force exerted by the axe on the tree upon impact is given by:
Force exerted by the axe = -Force exerted by the tree
Therefore, the force exerted by the axe on the tree upon impact is equal in magnitude but opposite in direction to the force exerted by the tree on the axe head.