On passing through a wooden sheet a bullet looses 1/20 f initial velocity.the minimum number of sheets required to completely stop bullet will be?

To determine the minimum number of wooden sheets required to completely stop a bullet that loses 1/20 of its initial velocity upon passing through each sheet, we can break down the problem step by step. This involves understanding how the bullet's velocity decreases with each sheet it penetrates.

Understanding Velocity Loss

When the bullet passes through a wooden sheet, it loses a fraction of its velocity. Specifically, it retains 19/20 of its velocity after passing through one sheet. If we denote the initial velocity of the bullet as V, after passing through one sheet, the velocity becomes:

V₁ = V × (19/20)

Subsequent Sheets

For each additional sheet, the bullet continues to lose velocity in the same manner. After passing through the second sheet, the velocity will be:

V₂ = V₁ × (19/20) = V × (19/20)²

Continuing this pattern, after passing through n sheets, the velocity can be expressed as:

V_n = V × (19/20)ⁿ

Condition for Stopping the Bullet

The bullet is considered to be completely stopped when its velocity becomes zero or effectively negligible. In practical terms, we can set a threshold where the velocity is less than or equal to zero. However, since we are dealing with a fraction, we can consider the bullet to be stopped when its velocity is less than a very small value, say ε (epsilon). Thus, we can set up the inequality:

V × (19/20)ⁿ ≤ ε

Finding the Minimum Number of Sheets

To find the minimum number of sheets n required to stop the bullet, we can rearrange the inequality:

(19/20)ⁿ ≤ ε/V

Taking the logarithm of both sides gives us:

n × log(19/20) ≤ log(ε/V)

Since log(19/20) is a negative number, we can flip the inequality when dividing:

n ≥ log(ε/V) / log(19/20)

Calculating with Specific Values

To find a numerical answer, we need to choose a small value for ε. For example, if we take ε = 0.01V, we can substitute this into our equation:

n ≥ log(0.01) / log(19/20)

Calculating these logarithms (using a calculator or logarithm tables):

• log(0.01) = -2
• log(19/20) ≈ -0.0223

Now substituting these values:

n ≥ -2 / -0.0223 ≈ 89.64

Final Result

Since we cannot have a fraction of a sheet, we round up to the nearest whole number. Therefore, the minimum number of wooden sheets required to completely stop the bullet is:

90 sheets

This calculation shows how the bullet's velocity diminishes with each sheet, and by applying logarithmic functions, we can determine the exact number of sheets needed for complete deceleration. Understanding these principles can be quite useful in physics, especially in topics related to motion and energy loss.