A bullet is shot through two cardboard disks attached a distance – Free 84A

A bullet is shot through two cardboard disks attached a distance D apart to a shaft turning with a rotational period T, as shown. Derive a formula for the bullet speed v in terms of D, T, and a measured angle ? between the position of the hole in the first disk and that of the hole in the second. If required, use ?, not its numeric equivalent. Both of the holes lie at the same radial distance from the shaft. ? measures the angular displacement between the two holes; for instance, ?=0 means that the holes are in a line and ?=? means that when one hole is up, the other is down. Assume that the bullet must travel through the set of disks within a single revolution.

Answer

Deriving the Bullet Speed Formula Using Rotational Motion and Angular Displacement

Understanding the physics behind motion through rotating systems can reveal fascinating relationships between time, distance, and angular motion. Let’s derive a mathematical expression for the speed of a bullet as it passes through two rotating disks, which are part of a shaft spinning with constant angular velocity.

🔍 Problem Overview

We have:

• D = distance between the two cardboard disks (in meters)
• T = rotational period of the shaft (in seconds)
• θ = angular displacement between the two holes (in radians)

The goal is to find the bullet’s linear speed v in terms of D, T, and θ.

🧠 Step 1: Understand the Motion

When the bullet passes through the first disk, it makes a hole. As it travels toward the second disk, the shaft continues to rotate. When the bullet reaches the second disk, the shaft has rotated by an angular displacement of θ radians.

🕰 Step 2: Time for Angular Displacement

The shaft completes one full rotation (2π radians) in a time period of T seconds. So the angular speed ω (omega) of the shaft is:

ω = 2π / T (radians per second)

The time t taken by the bullet to travel from the first to the second disk corresponds to the angular displacement θ:

t = θ / ω

Substitute ω = 2π / T:

t = θ × (T / 2π)

📏 Step 3: Use Linear Speed Formula

Speed is distance over time. The bullet travels a linear distance D in time t, so its speed v is:

v = D / t

Substitute the expression for t:

v = D / (θ × T / 2π)

✅ Final Bullet Speed Formula

Simplify the expression:

v = (2πD) / (θT)

🧪 Interpretation of the Formula

• As θ increases, meaning the angular gap between the holes widens, the bullet took more time to reach the second disk → speed v is lower.
• If T is larger (slower rotation), the bullet takes more time → v is smaller.
• If D increases (larger gap between disks), the bullet must move faster to create the same angular shift.

📘 Real-World Insight

This kind of setup is used in high-speed experiments to measure bullet velocities when digital sensors are not viable. It combines rotational kinematics and linear motion—a perfect harmony of physics concepts!