A 26 bullet strikes and becomes embedded in a 1.35 block of wood placed on a horizontal surface just in front of the gun. If the coefficient of kinetic friction between the block and the surface is 0.23, and the impact drives the block a distance of 9.9 before it comes to rest, what was the muzzle speed of the bullet?

There will be two parts to this question.

1) the bullet strikes and stays inside the wood
2) both bullet and wood block travel as as a system for 9.9m, under the friction of 0.23.

Mass of bullet= Mb, Vb=speed of bullet
Mass of wood bloc=Mw, Vw= speed of wood
Mass of bullet and wood block=Ms, Vs= speed of system

Step 1:
1) MaVa+ MwVw= (Ma+Mw) Vs; Vw at rest initially.
MaVa=(Ma+Mw)Vs

Step 2
1) Friction force= u.Ms.g= 0.23x1.376x9.8 =3.10N
Friction work= F.distance= 3.10N x 9.9m=30.7J

Step 3
(we need to calculate the Kinetic energy before and after of the system, as it travels actoss 9.9m with friction.

1) KEinitial= KE final
Work friction+ KE initial = KE final
Wf + 0.5MsVsinital(squared) 0.5MsVsfinal
((Vs final is zero after 9.9m. So we're left))
Wf + O.5MsVsinitial(squared )=0
Wf= 0.5MsVsinitial(squared)
Vsinitial(squared)= 2Wf/Ms
= 2(30.7J)/1.376kg
Vsinitial= 6.68m/s

Step 4
(plug the Vs initial answer from step 3 into the Vs in step 1)

MaVa=(Ma+Mw)6.68m/s
Va=(Ma+Mw)6.68/Ma
=(1.376x 6.68/ 0.026kg
= 353.5m/s <- (ANSWER)

To determine the muzzle speed of the bullet, we can use the principle of conservation of energy.

Step 1: Calculate the work done by the friction force.
The work done by the friction force is equal to the force of friction multiplied by the distance over which it acts. In this case, the force of friction can be calculated using the coefficient of kinetic friction (μ) and the normal force (N).

The normal force (N) can be calculated using the weight of the block (mg), where m is the mass of the block and g is the acceleration due to gravity.

Given:
Coefficient of kinetic friction (μ) = 0.23
Mass of the block (m) = 1.35 kg
Acceleration due to gravity (g) ≈ 9.8 m/s^2

Normal force (N) = mg = (1.35 kg)(9.8 m/s^2) = 13.23 N

The force of friction (F_friction) = μN = (0.23)(13.23 N) = 3.04 N

The work done by the friction force (W_friction) = F_friction × distance = (3.04 N)(9.9 m) = 30.096 J

Step 2: Calculate the initial kinetic energy of the block.
The initial kinetic energy of the block can be calculated using the work-energy theorem. Since the block starts from rest, the initial kinetic energy is zero.

Initial kinetic energy (K_i) = 0 J

Step 3: Calculate the final kinetic energy of the block.
The final kinetic energy of the block is also zero since it comes to rest.

Final kinetic energy (K_f) = 0 J

Step 4: Apply the principle of conservation of energy.
According to the principle of conservation of energy, the change in energy is equal to the work done on the block.

Change in kinetic energy (ΔK) = K_f - K_i = 0 J - 0 J = 0 J

The work done on the block (W) is equal to the work done by the friction force (W_friction).

Therefore, ΔK = W = 30.096 J

Step 5: Calculate the initial kinetic energy of the bullet.
The initial kinetic energy of the bullet can be calculated using the mass of the bullet (m_bullet) and the muzzle velocity (v_bullet).

Given:
Mass of the bullet (m_bullet) = 26 g = 0.026 kg

Initial kinetic energy of the bullet (K_bullet) = 1/2 m_bullet v_bullet^2

Step 6: Calculate the muzzle velocity of the bullet.
Using the equation ΔK = W, we can set the initial kinetic energy of the bullet equal to the work done on the block.

0.5(0.026 kg)v_bullet^2 = 30.096 J

Simplifying the equation, we find:

v_bullet^2 = (2 * 30.096 J) / 0.026 kg
v_bullet^2 = 2314.1536

Taking the square root of both sides, we find:

v_bullet ≈ 48.109 m/s

Therefore, the muzzle speed of the bullet is approximately 48.109 m/s.

To find the muzzle speed of the bullet, we can use the concept of work-energy theorem.

The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done is the force exerted by friction multiplied by the distance the block moves. The change in kinetic energy is the difference between the initial kinetic energy (which is the energy the bullet carries) and the final kinetic energy (which is zero when the block comes to rest).

Let's break down the steps to find the muzzle speed of the bullet:

1. Identify the given information:
- Mass of the bullet (m): 26 g = 0.026 kg
- Mass of the block (M): 1.35 kg
- Coefficient of kinetic friction (μ): 0.23
- Distance the block moves (d): 9.9 m

2. Calculate the work done:
The work done is given by the formula:
Work = Force × Distance

The force of friction (F) can be determined using the equation:
F = μ × Normal force

The normal force (N) is the weight of the block and can be calculated as:
N = M × g
where g is the acceleration due to gravity (approximately 9.8 m/s²).

After obtaining the force, multiply it by the distance to get the work done.

3. Calculate the change in kinetic energy:
The initial kinetic energy (K1) of the system (bullet and block) can be calculated as:
K1 = 0.5 × M × (velocity of the block)²
As the block moves with the bullet, their velocities are the same.

Since the final kinetic energy (K2) is zero when the block comes to rest, the change in kinetic energy is given by:
ΔK = K2 - K1 = -K1

4. Apply the work-energy theorem:
According to the work-energy theorem:
Work = ΔK

Set the work done by friction equal to the change in kinetic energy and solve for the unknown velocity (muzzle speed).

5. Convert units:
Make sure to convert the bullet mass to kilograms and the distance to meters, if they are given in different units.

By following these steps, you should be able to solve for the muzzle speed of the bullet.