Final answer:
The penetration distance of a bullet into a plank can be calculated using the proportion of kinetic energies based on the bullet's velocity. If a bullet penetrates a plank 5 cm at 100 m/s and the velocity increases to 300 m/s, the new distance penetrated would be 45 cm, based on kinetic energy being proportional to the square of the velocity.
Step-by-step explanation:
To solve this problem, we can make use of the kinetic energy and work principle in physics. When a bullet penetrates a plank, the work done to stop the bullet is equal to its initial kinetic energy. This work can be expressed as the force exerted by the plank on the bullet times the distance the bullet travels into the plank. Assuming the force exerted by the plank is constant, we can set up a proportion based on the kinetic energy of the bullet and the distance it travels.
When the bullet strikes at 100 m/s and penetrates 5 cm (0.05 m), we can assume it has certain kinetic energy (KE1). If the bullet strikes at 300 m/s, its kinetic energy (KE2) will be significantly higher. Specifically, kinetic energy is proportional to the square of the velocity, thus KE2 will be nine times KE1 because (300 m/s)^2 is nine times larger than (100 m/s)^2.
Now, if we assume the resistance force the plank exerts is constant, the distance penetrated (d2) when the bullet strikes at 300 m/s can be predicted by the ratio of the kinetic energies, since KE is also equal to the work done (Work = Force × Distance), and if Force remains constant, then the Distance is directly proportional to the KE:
KE1 is associated with d1 = 0.05 m.
KE2 is 9 × KE1 because (300 / 100)^2 = 9.
Therefore, d2 should be 9 × d1 = 9 × 0.05 m = 0.45 m or 45 cm.
Hence, if the bullet's initial velocity is increased to 300 m/s, it would penetrate the plank a distance of 45 cm, assuming a constant resistance force by the plank.