(a) Suppose the speed of light were only \(3000 \mathrm{m} / \mathrm{s}\). A jet fighter moving toward a target on the ground at 800 \(\mathrm{m} / \mathrm{s}\) shoots bullets, each having a muzzle velocity of 1000 \(\mathrm{m} / \mathrm{s} .\) What are the bullets' velocity relative to the target? (b) If the speed of light was this small, would you observe relativistic effects in everyday life? Discuss.

The velocity addition formula is crucial when discussing motion in physics, especially in scenarios involving high speeds. In classical mechanics, when you want to find the velocity of an object relative to a third party, you simply add their velocities relative to each other.

For example, if a jet moves towards a target at 800 m/s and shoots a bullet at a muzzle velocity of 1000 m/s, the bullet's velocity relative to the target is found by adding the jet's speed to the bullet's muzzle velocity. Thus, the bullet's velocity relative to the target (\(v_{BG}\)) is calculated as \(v_{BG} = v_{BJ} + v_{JG}\), where \(v_{BJ}\) is the muzzle velocity of the bullet, and \(v_{JG}\) is the velocity of the jet relative to the ground.

However, this formula changes slightly when objects move at speeds that are a significant fraction of the speed of light which brings in the concept of relativistic velocity addition.

The speed of light in a vacuum, denoted by 'c,' is a fundamental constant of nature that has a value of approximately 299,792,458 m/s. It is the maximum speed at which all energy, matter, and information in the universe can travel.

It's significant because it not only defines the speed of light but also appears in Einstein's famous equation \(E=mc^2\), which describes the relationship between mass and energy. If, hypothetically, the speed of light were much slower, as in this textbook exercise where it's suggested to be 3000 m/s, the laws of physics as we know them would be vastly different, altering the very fabric of how our universe functions. This would make relativistic effects much more apparent in our day-to-day experiences.

Muzzle velocity refers to the speed at which a bullet or projectile exits the barrel of a firearm. It is a key factor in determining a bullet's trajectory and hitting power. In our exercise, the muzzle velocity is given as 1000 m/s, which is the speed of the bullet relative to the jet.

The muzzle velocity is directly influenced by factors such as the gunpowder's quality, the firearm's design, and the mass of the bullet. Higher muzzle velocities typically allow for a flatter shot and a quicker reach to the target, but they can also be accompanied by increased recoil and wear on the firearm.

In physics, a reference frame is a perspective from which you measure and define movement. There are two types: inertial, with no accelerating forces, and non-inertial, which are accelerating or rotating.

Whether an object is moving or not, and how fast, depends on the observer's reference frame. In our exercise, there are two main reference frames: one from the jet's perspective and one from the ground target. The motion of the bullet is measured differently depending on the reference frame, demonstrating the relative nature of velocity. In all reference frames, though, the laws of physics hold true, solidifying the concept as a keystone of both classical and modern physics.