91Ó°ÊÓ

Muzzle speed refers to the velocity at which a bullet leaves the barrel of a gun upon firing. It is an important factor in determining the distance and impact force the bullet will have. In our exercise, the muzzle speed is given as \(150 \text{ m/s}\). This is the speed of the bullet relative to the policeman firing the weapon, not the speed relative to the ground or any other object.The muzzle speed can be influenced by a few factors:

• The type of firearm used.
• The design of the ammunition.
• Environmental factors like wind resistance.

Understanding muzzle speed is crucial for calculations involving moving targets, especially in scenarios like a police chase, where precision is necessary.

Unit conversion is a key step in solving problems involving speed and velocity, ensuring consistency and accuracy. In this exercise, speeds were initially given in kilometers per hour (km/h), which needed to be converted to meters per second (m/s) since muzzle speed is provided in m/s.The conversion from km/h to m/s is done using the formula:\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{1000}{3600} \]This effectively converts because 1 kilometer equals 1000 meters, and 1 hour equals 3600 seconds. For example, the policeman's speed of \(30 \text{ km/h}\) is converted to approximately \(8.33 \text{ m/s}\). Such conversions are critical to ensure that all velocities are compared on a like-for-like basis.

Once we understand the muzzle speed, we need to determine the speed of the bullet with respect to the ground. This involves combining the speed of the moving policeman with the muzzle speed of the bullet. The bullet's speed relative to the ground is calculated by adding the policeman's ground speed to the muzzle speed:\[ \text{Bullet Speed (ground)} = \text{Muzzle Speed} + \text{Policeman's Speed} \]In the given exercise, this results in:\[ 158.33 \text{ m/s} = 150 \text{ m/s} + 8.33 \text{ m/s} \]This calculation gives us the bullet's velocity as it moves through the air, providing a solid foundation for further calculations involving varying reference frames.

Ground speed is crucial in understanding how different velocities interact in a moving system. It is the speed of an object relative to a fixed point on the earth. In our case, it's the speed of the bullet as seen from the ground.In scenarios where everything moves in relation to something else, such as vehicles on a highway, understanding ground speed helps in:

• Determining real-world impacts.
• Calculating the outcome of interactions between objects (such as bullet and car in this case).

The relative speed of the bullet to the thief's car involves subtracting the car's ground speed from the bullet's ground speed:\[ \text{Bullet's Speed (relative to car)} = \text{Bullet Speed (ground)} - \text{Car Speed (ground)} \]\[ 105 \text{ m/s} = 158.33 \text{ m/s} - 53.33 \text{ m/s} \]This final result tells us how fast the bullet is moving towards the car, completing the puzzle of relative motion.