a bullet fired into a large block of wood of uniform resistance,hits the block with a velocity of 48ms^-1.it penetrates a distance of 60cm and the velocity is then reduced to 24ms^-1.it then penetrates a further distance of

To analyze the situation of a bullet penetrating a block of wood, we can apply the principles of physics, particularly the concepts of deceleration and resistance. Let's break down the problem step by step to understand how far the bullet penetrates after its velocity is reduced.

Understanding the Initial Conditions

The bullet initially travels at a velocity of 48 m/s and penetrates the wood to a depth of 60 cm (or 0.6 m) before its speed decreases to 24 m/s. We need to determine how much further the bullet will penetrate after reaching this new speed.

Applying the Work-Energy Principle

According to the work-energy principle, the work done on the bullet by the resistance of the wood is equal to the change in kinetic energy of the bullet. The kinetic energy (KE) of an object is given by the formula:

• KE = 0.5 * m * v²

Where m is the mass of the bullet and v is its velocity. Let's denote:

• Initial velocity (v₁) = 48 m/s
• Final velocity (v₂) = 24 m/s
• Initial penetration depth (d₁) = 0.6 m

Calculating the Change in Kinetic Energy

First, we calculate the initial and final kinetic energies:

• Initial KE (KE₁) = 0.5 * m * (48)² = 1152m
• Final KE (KE₂) = 0.5 * m * (24)² = 288m

The change in kinetic energy (ΔKE) as the bullet penetrates the wood is:

• ΔKE = KE₁ - KE₂ = 1152m - 288m = 864m

Relating Work Done to Penetration

The work done by the resistance of the wood (W) can also be expressed as:

• W = F * d

Where F is the average resistive force exerted by the wood and d is the distance penetrated. Since the resistive force is uniform, we can assume it remains constant throughout the penetration. The work done during the first penetration (d₁) is equal to the change in kinetic energy:

• F * d₁ = 864m

Finding the Resistive Force

From the above equation, we can express the resistive force:

• F = 864m / 0.6 = 1440m

Calculating Further Penetration

Now, we need to find out how far the bullet penetrates after its speed reduces to 24 m/s. The remaining kinetic energy at this point is:

• KE₂ = 288m

Setting this equal to the work done during the further penetration (d₂):

• F * d₂ = 288m

Substituting the value of F:

• 1440m * d₂ = 288m

Solving for d₂ gives:

• d₂ = 288m / 1440m = 0.2 m

Final Results

The bullet will penetrate an additional distance of 0.2 m, or 20 cm, after its velocity is reduced to 24 m/s. Therefore, the total penetration distance of the bullet into the wood is:

• Total penetration = d₁ + d₂ = 0.6 m + 0.2 m = 0.8 m

In summary, the bullet penetrates a total distance of 80 cm into the block of wood before coming to a stop. This example illustrates how the principles of energy and force interact in real-world scenarios, providing a clear understanding of the bullet's behavior as it moves through a resistive medium.