Answer:
Step-by-step explanation:
A. To find the average frictional force that stops the bullet, we can use the principle of conservation of energy. Initially, the bullet has kinetic energy given by:
KE = (1/2)mv^2
where m is the mass of the bullet and v is its velocity. The bullet stops when all of its kinetic energy is converted to thermal energy due to friction with the tree. We can calculate the thermal energy using the equation:
Q = cmΔT
where Q is the thermal energy, c is the specific heat capacity of the bullet (assumed to be that of lead, 0.128 J/g·K), m is the mass of the bullet, and ΔT is the change in temperature. We assume that all of the thermal energy is absorbed by the bullet, so ΔT is the change in temperature of the bullet.
The work done by the frictional force on the bullet is equal to the initial kinetic energy of the bullet, so we can write:
W = Fd = KE
where W is the work done by the frictional force, F is the average frictional force, and d is the distance over which the bullet is stopped (the depth of penetration of the bullet).
We can use the above equations to solve for the average frictional force:
KE = (1/2)mv^2 = W = Fd
F = KE/d = (1/2)mv^2/d
Substituting the given values, we get:
F = (1/2)(6.89 g)(581 m/s)^2 / (4.83 cm) = 1.91 N
Therefore, the average frictional force that stops the bullet is 1.91 N.
B. To find the time elapsed between the moment the bullet entered the tree and the moment it stopped, we can use the equation of motion:
d = (1/2)at^2
where d is the distance over which the bullet is stopped (the depth of penetration of the bullet), a is the acceleration of the bullet due to the frictional force (assumed to be constant), and t is the time elapsed between the moment the bullet entered the tree and the moment it stopped.
We can rearrange the equation to solve for t:
t = sqrt(2d/a)
Substituting the given values, we get:
t = sqrt(2(4.83 cm) / (1/2)(F/m))
t = sqrt(9.66 cm^2·g / N)
t = 0.174 s
Therefore, the time elapsed between the moment the bullet entered the tree and the moment it stopped is 0.174 s.