Understanding the Ballistics Problem

This physics problem deals with the concepts of kinetic energy and its change when a bullet passes through a material like soft plywood. We are given the initial mass and speed of the bullet, the thickness of the plywood (though this information is not needed for the calculation of emergent speed based on energy loss), and the percentage of initial kinetic energy the bullet retains after passing through.

The key is to use the kinetic energy formula to find the final speed of the bullet after it has lost most of its initial energy.

Step-by-Step Calculation of Emergent Speed

Let's break down the problem and calculate the emergent speed:

1. Identify the given quantities:
   • Mass of the bullet, $m = 60.0 \text{ g}$
   • Initial speed of the bullet, $v_{initial} = 200 \text{ ms}^{-1}$
   • Thickness of plywood = $4.00 \text{ cm}$ (This is extra information not directly used here)
   • Final kinetic energy is 8% of initial kinetic energy.
2. Convert units:
   • Mass needs to be in kilograms for standard energy calculations: $m = 60.0 \text{ g} = \frac{60.0}{1000} \text{ kg} = 0.060 \text{ kg}$.
3. Calculate the initial kinetic energy:
   • The formula for kinetic energy (KE) is $\text{KE} = \frac{1}{2}mv^2$.
   • Initial kinetic energy, $\text{KE}_{initial} = \frac{1}{2} \times m \times v_{initial}^2$
   • $\text{KE}_{initial} = \frac{1}{2} \times 0.060 \text{ kg} \times (200 \text{ ms}^{-1})^2$
   • $\text{KE}_{initial} = 0.030 \text{ kg} \times (40000 \text{ m}^2\text{s}^{-2})$
   • $\text{KE}_{initial} = 1200 \text{ J}$
4. Calculate the final kinetic energy:
   • The bullet emerges with 8% of its initial kinetic energy.
   • Final kinetic energy, $\text{KE}_{final} = 8\% \text{ of } \text{KE}_{initial}$
   • $\text{KE}_{final} = 0.08 \times 1200 \text{ J}$
   • $\text{KE}_{final} = 96 \text{ J}$
5. Calculate the emergent speed (final speed):
   • We know the final kinetic energy $\text{KE}_{final}$ and the mass $m$. We can use the kinetic energy formula again to find the final speed, $v_{final}$.
   • $\text{KE}_{final} = \frac{1}{2}m v_{final}^2$
   • $96 \text{ J} = \frac{1}{2} \times 0.060 \text{ kg} \times v_{final}^2$
   • $96 = 0.030 \times v_{final}^2$
   • Rearrange to solve for $v_{final}^2$: $v_{final}^2 = \frac{96}{0.030}$
   • $v_{final}^2 = 3200$
   • Take the square root to find $v_{final}$: $v_{final} = \sqrt{3200}$
   • $v_{final} \approx 56.5685 \text{ ms}^{-1}$

Comparing this calculated value to the given options, 56.5685 ms⁻¹ is closest to 56.55 ms⁻¹.

Conclusion on Bullet Emergent Speed

By calculating the initial kinetic energy and then determining the final kinetic energy (8% of the initial), we were able to use the kinetic energy formula in reverse to find the emergent speed of the bullet. The emergent speed is approximately 56.57 ms⁻¹, which matches one of the provided options closely.

Revision Table: Key Concepts

Additional Information: Energy Loss in Ballistics

When a bullet penetrates a material like plywood, it loses energy primarily due to the work done against resistive forces within the material. These forces can include friction, deformation of the material, and breaking of bonds within the material structure.

• The energy lost by the bullet is converted into other forms, such as heat in the plywood and the bullet, sound, and energy used to break the material apart.
• In this specific problem, we weren't asked to calculate the work done by the resistive forces, but we could find it by subtracting the final kinetic energy from the initial kinetic energy ($W_{resistive} = \text{KE}_{initial} - \text{KE}_{final}$). This energy loss represents the work done by the plywood on the bullet.
• The thickness of the plywood (4.00 cm) might be relevant if we were asked to calculate the average resistive force, but it is not needed to find the emergent speed if the final kinetic energy is given as a percentage of the initial.