A bullet losses ` 1/6 `of its velocity in passing through a plank. What is the least number of planks required to stop the bullet ?

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem of how many planks are required to stop a bullet that loses \( \frac{1}{6} \) of its velocity after passing through each plank, we can follow these steps:

### Step 1: Determine the final velocity after passing through one plank
The bullet loses \( \frac{1}{6} \) of its velocity when it passes through a plank. If the initial velocity of the bullet is \( u \), then the velocity after passing through one plank, \( v \), can be calculated as:

\[ v = u - \frac{u}{6} = u \left(1 - \frac{1}{6}\right) = u \cdot \frac{5}{6} \]

### Step 2: Calculate the change in velocity
The change in velocity \( \Delta v \) after passing through one plank is:

\[ \Delta v = u - v = u - \frac{5u}{6} = \frac{u}{6} \]

### Step 3: Determine the total number of planks needed to stop the bullet
Let \( n \) be the number of planks required to stop the bullet. After passing through \( n \) planks, the bullet will have lost a total velocity of \( n \cdot \Delta v \). The bullet will come to rest when its final velocity \( v_n = 0 \). Therefore, we can set up the equation:

\[ u - n \cdot \frac{u}{6} = 0 \]

### Step 4: Solve for \( n \)
Rearranging the equation gives:

\[ n \cdot \frac{u}{6} = u \]

Dividing both sides by \( u \) (assuming \( u \neq 0 \)):

\[ n \cdot \frac{1}{6} = 1 \]

Multiplying both sides by 6:

\[ n = 6 \]

### Step 5: Conclusion
The least number of planks required to stop the bullet is \( n = 6 \).