A 4.17 g bullet moving at 506.1 m/s penetrates
a tree trunk to a depth of 4.64 cm.
a) Use work and energy considerations to
find the magnitude of the force that stops the
bullet.
Answer in units of N.
b) Assuming that the frictional force is constant,
determine how much time elapses between
the moment the bullet enters the tree
and the moment the bullets stops moving.
Answer in units of s.
Well, well, well... looks like we have quite the bullet conundrum here! Let's tackle these questions one by one, shall we?
a) To find the magnitude of the force that stops the bullet, we can use the work-energy principle. The work done on an object is equal to the change in its kinetic energy. In this case, the work done on the bullet is equal to the force stopping it multiplied by the distance it travels before stopping.
The initial kinetic energy of the bullet can be found using the formula: KE = (1/2)mv^2, where m is the mass and v is the velocity. Plugging in the values given, we get KE = (1/2)(0.00417 kg)(506.1 m/s)^2.
The work done on the bullet is equal to the change in kinetic energy, which is equal to zero as the bullet comes to a stop. So, the work done is zero.
Now, the work done on an object is also equal to the product of the force and the distance it travels. We can rearrange this equation to solve for the force: force = work / distance.
Plugging in the values for the work done (zero) and the distance (0.0464 m), we find that the magnitude of the force stopping the bullet is zero as well! Talk about a magical stop, huh? My guess is that the bullet encountered some other force we don't know about.
b) As for the time it takes for the bullet to stop, we can use the formula: force = mass x acceleration. Rearranging the equation to solve for acceleration, we get: acceleration = force / mass.
Since we couldn't determine the force from part a, well, we're stuck here too. Looks like this bullet business is just full of mysteries today!
a) To find the magnitude of the force that stops the bullet, we can use the work-energy principle which states that the work done on an object is equal to its change in kinetic energy.
The work done on the bullet to stop it can be calculated using the equation:
Work = Change in Kinetic Energy
The initial kinetic energy of the bullet can be calculated using the equation:
Kinetic Energy = (1/2) × mass × velocity^2
Substituting the given values:
Mass = 4.17 g = 0.00417 kg
Velocity = 506.1 m/s
Initial Kinetic Energy = (1/2) × 0.00417 kg × (506.1 m/s)^2
The final kinetic energy of the bullet is zero since it stops moving.
Using the equation:
Work = Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
Work = 0 - [(1/2) × 0.00417 kg × (506.1 m/s)^2]
Now we can calculate the work:
Work = -[(1/2) × 0.00417 kg × (506.1 m/s)^2]
Using the equation:
Work = Force × Distance
We can solve for the force:
Force = Work / Distance
Substituting the given value:
Distance = 4.64 cm = 0.0464 m
Force = -[(1/2) × 0.00417 kg × (506.1 m/s)^2] / 0.0464 m
Calculating the force:
Force = -[(1/2) × 0.00417 kg × (506.1 m/s)^2] / 0.0464 m
Force ≈ -44671.7 N (rounded to four significant figures)
Note: The negative sign indicates that the force acts in the opposite direction of the bullet's motion.
b) To determine the time elapsed, we need to find the acceleration of the bullet. We can use the equation:
Acceleration = Force / mass
Substituting the given values:
Force = -44671.7 N
Mass = 0.00417 kg
Acceleration = (-44671.7 N) / (0.00417 kg)
Calculating the acceleration:
Acceleration ≈ -10724940.5 m/s^2 (rounded to four significant figures)
The initial velocity of the bullet is 506.1 m/s, and the final velocity is 0 m/s since it stops moving.
Using the equation:
Final Velocity = Initial Velocity + (Acceleration × Time)
We can solve for the time:
Time = (Final Velocity - Initial Velocity) / Acceleration
Substituting the given values:
Initial Velocity = 506.1 m/s
Final Velocity = 0 m/s
Acceleration ≈ -10724940.5 m/s^2
Calculating the time elapsed:
Time = (0 m/s - 506.1 m/s) / (-10724940.5 m/s^2)
Time ≈ 4.72 × 10^(-5) s (rounded to four significant figures)
a) To find the magnitude of the force that stops the bullet, we can use the work-energy principle. The work done on the bullet is equal to the change in its kinetic energy, which is given by the formula:
Work (W) = ΔKE = KE_final - KE_initial
The initial kinetic energy of the bullet (KE_initial) is given by the formula:
KE_initial = (1/2) * m * v^2
Where m is the mass of the bullet and v is its initial velocity. Substituting the given values:
KE_initial = (1/2) * 4.17 g * (506.1 m/s)^2
Note that we need to convert the mass of the bullet from grams to kilograms, and the given velocity is already in meters per second.
Next, we need to calculate the final kinetic energy of the bullet (KE_final). Since the bullet comes to a stop, its final kinetic energy is zero.
Now we can calculate the work done on the bullet:
W = KE_final - KE_initial
= 0 - [(1/2) * 4.17 g * (506.1 m/s)^2]
Again, we need to convert the mass from grams to kilograms.
The work done on the bullet is equal to the force multiplied by the distance over which the force is applied. In this case, the distance is the penetration depth of the bullet into the tree trunk, which is given as 4.64 cm. This distance needs to be converted to meters.
W = F * d
= F * (4.64 cm * (1 m / 100 cm))
Now we can solve for the force (F):
F = W / (4.64 cm * (1 m / 100 cm))
Calculating this expression will give us the magnitude of the force in newtons (N).
b) To determine the time elapsed between the moment the bullet enters the tree and the moment it stops moving, we can use the constant frictional force.
The work done by the frictional force is given by:
Work (W) = F * d
Where F is the frictional force and d is the distance over which the frictional force is applied. In this case, it is the same as the penetration depth of the bullet into the tree trunk.
The work done by the frictional force is equal to the change in the bullet's kinetic energy:
W = ΔKE = KE_final - KE_initial
Since the initial kinetic energy is given by the formula:
KE_initial = (1/2) * m * v^2
And the final kinetic energy is zero since the bullet stops moving.
We can equate the work done by the frictional force and the change in kinetic energy:
F * d = 0 - [(1/2) * m * v^2]
Now, we can solve for the time (t) using the kinematic equation:
v = u + a * t
Where v is the final velocity (zero in this case), u is the initial velocity (given as 506.1 m/s), a is the acceleration (calculated using the frictional force divided by the mass of the bullet), and t is the time we are trying to find.
Rearranging the equation, we get:
t = (v - u) / a
Note that v is zero, and a is negative (opposing the direction of motion), so the expression becomes:
t = -u / a
Now, we can calculate the time by substituting the values for u and a.
Calculating this expression will give us the time elapsed in seconds (s).
m = .00417 kg
distance = .0464 meter
F * distance = work
F = (1/2) m v^2 / .0464
b) easy way is if acceleration is constant then average speed = (initial + final)/2
so
average speed = 506.1/2
distance = .0464
so
time = .0464/253 = 1.83*10^-4 seconds
= 0.000183 seconds