91Ó°ÊÓ

Angular displacement refers to the angle through which an object moves on a circular path. In the context of the bullet speed measurement, it's the angle at which the bullet holes appear to have shifted as the bullet passes through the two rotating disks. This angular measurement, denoted as \(\theta\), is crucial for determining the bullet's speed because it helps us understand how far along the rotation the bullet has moved during its travel from one disk to the next.

Think of angular displacement as the arc of a circle that gets traced out between two points rather than a straight line. In this problem, \(\theta = 0.240 \, \text{rad}\) tells us the angle between the two bullet marks due to the disk's rotation. Understanding this concept helps to relate linear motion (bullet's path) with rotational motion (disk's spinning).

Angular velocity, represented by \(\omega\), is the rate at which an object rotates or spins around a central point. In our case, it describes how fast the disks rotate. Angular velocity is typically measured in radians per second (\(\text{rad/s}\)). For the bullet speed measurement device, \(\omega = 95.0 \, \text{rad/s}\), indicating the speed at which the disks are spinning when the bullet passes through.

Understanding angular velocity is key for calculating the bullet's speed, because it allows us to find out how much time the disks took to rotate over the angle \(\theta\), while the bullet moved between them. This relay of information from rotational to linear is fundamental to solving the problem, making angular velocity an essential component.

Rotational motion is the movement of an object around a central axis or point. In this scenario, the disks are undergoing rotational motion. This motion is characterized by parameters such as angular velocity and angular displacement, which describe how fast and how far the disk rotates.

To solve the bullet speed problem, we leverage the rotational motion of the disks to determine the time elapsed between the bullet passing through the two disks. This motion helps bridge the gap between the rotational aspect (disks spinning) and the bullet's linear motion (straight-line path through the disks). By analyzing the rotational motion, we can convert angular information into a parameter needed for linear speed calculations, thereby linking the bullet's passage with the rotating disks.

Speed calculation involves determining how fast something is moving along its path. In this exercise, we are tasked to find the speed of the bullet using the rotational data from the disks. The key to calculation is using the time \(t\) it takes for the bullet to travel between the disks, which we've deduced through angular displacement and angular velocity.

The straightforward formula to find speed is \(v = \frac{d}{t}\), where \(d\) is the distance between the disks, and \(t\) is the time calculated from the rotational motion data. With \(d = 0.850 \, \text{m}\) and \(t = 0.002526 \, \text{s}\), the bullet's speed is calculated to be approximately \(336.48 \, \text{m/s}\). This speed calculation seamlessly integrates the rotational and linear motion concepts to give us the bullet’s actual speed as it travels from one disk to the next.