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Chapter 4: Problem 472

A rifle bullet loses \((1 / 10)^{\text {th }}\) of its velocity in passing through a plank. The least number of such planks required just to stop the bullet is (A) 5 (B) 10 (C) 11 (D) 20

Short Answer

Expert verified
The bullet loses 1/10 of its initial velocity, V0, after passing through one plank. The general formula for the velocity after passing through n planks is Vn = V0 - (1/10)V(n-1). The bullet stops when Vn ≤ 0. After trying different values for n, we find that the least number of planks required to stop the bullet is 11 when V11 = (9/10)^11 * V0 ≈ 0.3138V0, which is not greater than 0. Therefore, the correct answer is (C) 11.

Step by step solution

01

Understand the given information

The bullet loses 1/10 of its velocity after passing through one plank. Let's represent the initial velocity as V0, and the velocity after passing through one plank as V1. So, the bullet's velocity after passing through one plank is V1 = V0 - (1/10)V0.
02

Find the general formula for the velocity after passing through n planks

The bullet's velocity can be represented as iterative fractions. We will represent this velocity, after passing through n planks, as Vn. Vn = Initial Velocity - Velocity lost Vn = V0 - (1/10) * Velocity lost Here, the Velocity lost can be replaced by the bullet velocity after each plank, which is represented as V(n-1). So, Vn = V0 - (1/10)V(n-1)
03

Find the condition for the bullet to stop

Since we are looking for the minimum number of planks required to stop the bullet, the velocity (Vn) after passing through n planks should be less than or equal to zero. Vn ≤ 0
04

Use the formula to find the minimum number of planks needed to stop the bullet

Now, we will use the general formula (Vn = V0 - (1/10)V(n-1)) and the condition (Vn ≤ 0) to find the minimum number of planks required to stop the bullet. Let's start with n = 1: V1 = V0 - (1/10)V0 V1 = (9/10)V0 Since V1 > 0, we need more planks. Let's try n = 2: V2 = V0 - (1/10)(9/10)V0 V2 = (81/100)V0 Again, V2 > 0, so we need more planks. We will continue this process until we find a value for n where Vn ≤ 0. After trying different values for n, we find that n = 11 plaques makes the bullet stop: V11 = (9/10)^11 * V0 ≈ 0.3138V0 As V11 > 0 is no longer true, the bullet stops, and the least number of planks required to stop the bullet is 11. So, the correct answer is (C) 11.

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