A bullet is fired vertically into 20 identical wooden boards stacked together. The bullet comes to rest after penetrating the 20th board. We assume the bullet experiences the same deceleration in each board. Our goal is to find the time taken to pass through the 15th board, given that the total time in all boards is t.
To solve this problem, we need to apply kinematic equations. The bullet experiences constant deceleration in each board. We know that after passing through all 20 boards, the velocity becomes zero. The velocity decreases as the bullet penetrates more boards, following a specific mathematical relationship.
Let's solve this step by step. For uniform deceleration, we use the kinematic equation v squared equals v naught squared minus 2 a s. After n boards, the velocity squared is v naught squared minus 2 a n d. Since the bullet stops after 20 boards, we get v naught squared equals 40 a d. After 15 boards, the velocity squared becomes 10 a d.
To find the time spent in each board, we use the distance formula. Each board has thickness d, and the bullet enters with different velocities. The time for each board depends on the entry velocity. Through mathematical analysis, we find that the time for the 15th board is t over 20 times the quantity square root of 6 minus square root of 5.