A 7.80-g bullet moving at 530 m/s penetrates a tree trunk to a depth of 6.2 cm.
(a) Use work and energy considerations to find the average frictional force that stops the bullet.
(b) Assuming the friction force is constant, determine how much time elapses between the moment the bullet enters the tree and the moment it stops moving
change in KE =1/2mv^2
1/2(.0078)(530)^2=1095.51 kg*m^2/s^2
6.2 cm = .0062 m
1095.51/.0062 = 176,695 N= F
What formula did you use?
Why did you divide the KE with the depth of the penetration?
To solve this problem, we can use the work-energy principle, which states that the net work done on an object is equal to its change in kinetic energy.
(a) First, let's find the initial kinetic energy of the bullet.
Given:
- Mass of the bullet (m): 7.80 g = 0.00780 kg
- Initial velocity of the bullet (v): 530 m/s
The formula for kinetic energy (K.E.) is: K.E. = 0.5 * m * v^2
Plugging in the given values, we can calculate the initial kinetic energy of the bullet:
K.E. = 0.5 * 0.00780 kg * (530 m/s)^2
= 0.5 * 0.00780 kg * (280900 m^2/s^2)
= 0.5 * 0.00780 kg * 280900 J
≈ 1097.22 J
Now let's find the work done by the frictional force to bring the bullet to rest.
The work done by a force (W) is given by: W = F * d * cosθ
Given:
- Depth of penetration (d): 6.2 cm = 0.062 m (converted from centimeters to meters)
- Angle between the force and displacement (θ): 0 degrees (force and displacement are in the same direction)
We want to find the average frictional force (F).
Rearranging the formula, we get:
F = W / (d * cosθ)
We know that the net work done on the bullet is equal to the change in kinetic energy:
W = ΔK.E.
Since the bullet is brought to rest, the change in kinetic energy is equal to the negative of the initial kinetic energy:
ΔK.E. = -K.E.
Therefore, the work done by the frictional force is:
W = -K.E. = -1097.22 J
Plugging in the values, we can calculate the average frictional force:
F = -1097.22 J / (0.062 m * cos(0 degrees))
= -1097.22 J / (0.062 m)
Since the angle between the force and displacement is 0 degrees, the cosine of 0 degrees is 1.
So, the average frictional force that stops the bullet is approximately: F ≈ -17,693.23 N
(b) Now, let's determine the time it takes for the bullet to stop.
The formula for average force (F) is given by: F = m * a
Since the frictional force (F) is equal to the product of mass (m) and average acceleration (a), we can rearrange the formula to find the average acceleration:
a = F / m
Plugging in the values, we can calculate the average acceleration:
a = (-17,693.23 N) / 0.00780 kg
≈ -2,268,900 m/s^2
The bullet starts with an initial velocity (v) and comes to rest when its final velocity is 0. The average acceleration (a) is the change in velocity (Δv) divided by the time (t).
Using the formula Δv = a * t, we can rearrange it to find the time (t):
t = Δv / a
= (0 - 530 m/s) / (-2,268,900 m/s^2)
Simplifying the equation, we get:
t = 530 m/s / 2,268,900 m/s^2
Therefore, the time it takes for the bullet to stop moving is approximately: t ≈ 0.00023 seconds
To find the average frictional force that stops the bullet, we can use the work-energy principle. According to this principle, the work done on an object is equal to the change in its kinetic energy.
(a) The initial kinetic energy of the bullet can be calculated using the formula:
KE = (1/2)mv^2
where m is the mass of the bullet and v is its initial velocity. Given that the mass of the bullet is 7.80 g (or 0.00780 kg) and its velocity is 530 m/s, we can calculate the initial kinetic energy:
KE_initial = (1/2)(0.00780 kg)(530 m/s)^2
Next, we need to calculate the final kinetic energy of the bullet when it stops moving. Since the bullet comes to rest, its final velocity is 0 m/s. Therefore, the final kinetic energy is:
KE_final = (1/2)(0.00780 kg)(0 m/s)^2
Since kinetic energy is a scalar quantity, the change in kinetic energy (ΔKE) can be calculated as:
ΔKE = KE_final - KE_initial
Substituting the values, we find:
ΔKE = (1/2)(0.00780 kg)(0 m/s)^2 - (1/2)(0.00780 kg)(530 m/s)^2
The work done on the bullet to bring it to a stop is equal to the change in kinetic energy. So, we can use this value to find the average frictional force using the equation:
Work = Force × Distance
The distance is given as 6.2 cm, which we need to convert to meters:
Distance = 6.2 cm = 0.062 m
The work done on the bullet can be calculated as:
Work = Force × 0.062 m
Equating the work done to the change in kinetic energy, we have:
(1/2)(0.00780 kg)(0 m/s)^2 - (1/2)(0.00780 kg)(530 m/s)^2 = Force × 0.062 m
Now, solving for Force, we can find the average frictional force that stops the bullet.
(b) To determine the time elapsed between the moment the bullet enters the tree and the moment it stops moving, we can use the equation of motion:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. Since the bullet comes to rest, the final velocity is 0 m/s. The initial velocity is given as 530 m/s. We need to find the acceleration (a) and then determine the time (t).
The acceleration can be found using Newton's second law of motion:
F = ma
where F is the average frictional force calculated in part (a) and m is the mass of the bullet. Rearranging the equation, we have:
a = F / m
Substituting the values, we can calculate the acceleration.
Once we have the acceleration, we can use it along with the initial velocity to find the time elapsed using the equation of motion mentioned earlier:
0 m/s = 530 m/s + (acceleration)(t)
Solving for t, we can determine the time taken for the bullet to stop moving.