A bullet loses ` 1//20` of its velocity in passing through a plank. What is the least number of plank required to stop the bullet .

Options

A. 11
B. 20
C. 21
D. infinite

Answer

To solve the problem of determining the least number of planks required to stop a bullet that loses \( \frac{1}{20} \) of its velocity upon passing through each plank, we can follow these steps:

### Step 1: Determine the velocity after passing through one plank
Let the initial velocity of the bullet be \( u \). When the bullet passes through one plank, it loses \( \frac{1}{20} \) of its velocity. Thus, the velocity after passing through one plank, denoted as \( v \), can be calculated as follows:

\[ v = u - \frac{u}{20} = u \left(1 - \frac{1}{20}\right) = u \left(\frac{19}{20}\right) \]

### Step 2: Use the equation of motion
We can use the equation of motion to relate the initial and final velocities with acceleration and distance. The equation is:

\[ v^2 - u^2 = 2as \]

Where:
- \( v \) is the final velocity after passing through the plank,
- \( u \) is the initial velocity,
- \( a \) is the deceleration (negative acceleration),
- \( s \) is the distance traveled through the plank.

Substituting \( v = \frac{19u}{20} \) into the equation gives:

\[ \left(\frac{19u}{20}\right)^2 - u^2 = 2as \]

### Step 3: Simplify the equation
Calculating \( \left(\frac{19u}{20}\right)^2 \):

\[ \left(\frac{19u}{20}\right)^2 = \frac{361u^2}{400} \]

Now substituting back into the equation:

\[ \frac{361u^2}{400} - u^2 = 2as \]

Rearranging gives:

\[ \frac{361u^2}{400} - \frac{400u^2}{400} = 2as \]

This simplifies to:

\[ \frac{-39u^2}{400} = 2as \]

### Step 4: Relate the distance to multiple planks
Let \( n \) be the number of planks. The total distance \( s' \) traveled through \( n \) planks is:

\[ s' = n \cdot s \]

Substituting \( s' \) into the equation gives:

\[ 2a(n \cdot s) = \frac{-39u^2}{400} \]

### Step 5: Solve for \( n \)
We know that the bullet must come to a stop, which means the final velocity after passing through all \( n \) planks must be 0. Therefore, we can set up the equation:

\[ 0 - u^2 = 2a(n \cdot s) \]

Setting the two equations equal gives:

\[ 2a(n \cdot s) = \frac{-39u^2}{400} \]

Substituting the expression for \( 2as \) from earlier, we have:

\[ n \cdot \frac{-39u^2}{400} = -u^2 \]

Now, simplifying this gives:

\[ n \cdot \frac{39}{400} = 1 \]

Thus, solving for \( n \):

\[ n = \frac{400}{39} \approx 10.25 \]

### Step 6: Round up to the nearest whole number
Since \( n \) must be a whole number (you can't have a fraction of a plank), we round up to the next whole number:

\[ n = 11 \]

### Final Answer
The least number of planks required to stop the bullet is **11**.

---