Problem 472 - JEE Physics

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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)

Step 3: 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

Step 4: 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.

Key Concepts

Understanding Kinematics

Kinematics is the branch of physics that focuses on the motion of objects without considering the forces causing them. In this problem, the bullet travels through several planks, and its motion can be analyzed using kinematic concepts.

Key aspects to understand in kinematics include:

• Displacement - the change in position of the bullet.
• Velocity - how fast the bullet travels in a particular direction.
• Acceleration - the rate at which the velocity changes, often affected by forces like friction when passing through planks.

Here, our focus is primarily on velocity, especially the change in velocity as the bullet interacts with the planks.

Understanding Velocity Reduction

Velocity reduction refers to the decrease in the speed of the bullet as it interacts with an obstacle like a plank.

For this problem, we know that each plank reduces the bullet's velocity by \( \frac{1}{10} \). This implies each plank acts as a medium causing velocity loss.

• The bullet's velocity after passing one plank can be calculated using: \( V_1 = V_0 - (1/10)V_0 = (9/10)V_0 \).
• Velocity reduction accumulates with each subsequent plank.

Understanding this concept is essential for solving the problem – specifically finding out after how many planks the bullet's velocity becomes zero.

The Principles of Projectile Motion

Projectile motion involves objects moving in a curved path under the influence of gravity, after an initial force is applied. When applied to a bullet, it takes the form of a straight-line motion affected by resistance. While the traditional arc trajectory doesn’t apply here due to resistance from planks, key components of projectile motion include:

• Horizontal motion - the constant horizontal velocity component, which undergoes reduction here.
• Vertical motion - likely negligible here, but would normally include gravity’s impact.

In this problem, the focus is on continued horizontal interaction with the planks to reduce the bullet velocity to zero.

Effects of Motion Through Medium

Motion through a medium such as wood, in this case, impacts a bullet's velocity significantly. Each plank constitutes a medium that exerts resistive forces on the bullet.

• These resistive forces result in velocity reduction as analyzed earlier through mathematical calculations with each plank decreasing speed by \( \frac{1}{10} \).
• As the number of planks increases, these aggregate forces eventually bring the bullet to a complete stop, demonstrating a cumulative effect of resistance over multiple mediums.

Understanding motion through medium provides insights on the mechanics of friction and resistance, crucial for calculating how many planks are needed to stop the bullet.