Morning n' Morning n' Morning!

$v=\omega r = \frac{2\pi r}{T}=\frac{2\pi \times 6370\times 10^{3}}{24\times 60\times 60}= \boxed{463.24}m/s$ , to 2d.p.

To start off, I must first say that I often find that the solutions provided tend not to fully explain the process by which the answer was found. I've found that I best understand the concept being presented if I have the process explained to me step-by-step . For this purpose, my solution is lengthy and detailed , explaining the math and reasoning behind each step in the process. If you are like me and need step-by-step details , I hope you find my solution to be beneficial.

I thought of this scenario like a treadmill. In order for you to remain in the same place relative to your surroundings (the light fixtures, furniture, walls, etc.), you must be running at the same speed that the treadmill's track is moving . (Note that you must be running opposite the direction that the treadmill's track is moving.)

Time of day is basically a representation of the Sun's position in the sky, relative to where you are on the ground. That positioning changes as the Earth rotates, thus changing the portion of the Earth that receives sunlight. For the Sun to be in the same position relative to the jet (and thus the time remain the same relative to the jet), the jet must travel at the same velocity as the Earth's rotation. So, to solve this problem we must find the velocity of the Earth's rotation in meters per second .

We know that velocity is equal to distance per time: $v = \frac{d}{t}$ Let our distance d be the circumference of the Earth, $d = 2\pi r$ where r is the radius of the Earth, which is 6370 km . Let our time t be the time it takes the Earth to complete one full rotation, which is 24 hours . Note that we could use any distance on the Earth's surface and the time it takes to travel that distance, but the circumference and 24 hours are known values and are easy to work with. We will use them for simplicity.

Now that we know our variables, it's time to set up the problem.

We will first replace d in our velocity equation with our equation for the circumference of the Earth. $v = \frac{2\pi r}{t}$ Now let's replace the r and t with our radius and time variables, respectively. (Make sure to pay attention to the units , because they're important!) $v = \frac{2\pi\times6370km}{24hrs}$ We'll go ahead and simplify a little bit by reducing the fraction to make things cleaner. (Not necessary, but it helps me keep track of things more easily.) $v = \frac{\pi\times6370km}{12hrs}$ Now we have a problem. Our units are in km per hour , but we need them to be in meters per second . To fix this, we need to convert our units . We know that 1 km = 1000 m, 1 hr = 60 min, and 1 min = 60 seconds . Now if we substitute these values into our equation, we get $v = \frac{\pi\times6370\times1000m}{12\times60min} = \frac{\pi\times6370\times1000m}{12\times60\times60sec}$ Now that we have our units right, we can go ahead and use our calculator to solve our equation and find the answer! $v = \frac{\pi\times6370\times1000m}{12\times60\times60sec} = \boxed{463.24\frac{m}{sec}}$ Also note that the jet must be traveling west , which is opposite the direction of the Earth's rotation.