Clarification: this is calculus-based physics, so the methodology is not restricted to trigonometry only.

I have two questions, so hopefully I can divide them up nice and evenly.

Also, if you know where I should copy and paste proper vector notation from, please tell me so I can be more clear. Thanks!

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1. A plane flies 410 km east from city A to city B in 45.0 min and then 980 km south from city B to city C in 1.50 h. (Assume the +x axis points east and the +y axis points north.)
(a) In magnitude-angle notation, what is the plane's displacement for the total trip?
(b) In magnitude-angle notation, what is the plane's average velocity for the total trip?
(c) What is the plane's average speed for the trip?

(a) Let's forget about this, I got the correct answer.
(b) I'm applying the definition of average velocity into vectors, thus finding a displacement vector in the time interval and plugging in. Then I use the the standard formulas for magnitude and theta.
1) Δr=(BC)-(AB)=(-980km)j-(-410km)i=(-410km)i+(-980km)j.
2) Δt=1.50hr-0.750hr=0.750hr.
3) average velocity vector=(-547km/h)i+(-1.31e3km/h)j
4) |avg V vector|=√[(-547km/h)^2+(-1.31e3km/h)^2]=1.42e3km/h
5) θ=arctan[(-1.31e3km/h)/(-547km/h)]=67.3°
6) 180°+67.3°=247°
(c) Average speed is merely a magnitude, so just add the magnitude of each trip and divide by the time interval.
1) avg speed=(410km+980km)/0.75hr=1.85e3km/h

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2. An Earth satellite moves in a circular orbit 687 km above the Earth's surface. The period of the motion is 98.3 min.
(a) What is the speed of the satellite?
(b) What is the magnitude of the centripetal acceleration of the satellite?

(a) Using formula of T=2πr/v I solve for v. This yields v=2πr/T.
1) r=(mean earth radius)+(satellite height)=6,371,687m
2) v=(12,743,347m)π/5,898s=6.79e3m/s
(b) Now.. if the speed from (a) were correct, we could input into a=v^2/r for centripetal a. However, my calculated speed is incorrect :<

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Any insights on where I screwed up or should've looked at is greatly appreciated!

b) I'm applying the definition of average velocity into vectors, thus finding a displacement vector in the time interval and plugging in. Then I use the the standard formulas for magnitude and theta.

1) Δr=(BC)-(AB)=(-980km)j-(-410km)i=(-410km)i+(-980km)j.***correct
2) Δt=1.50hr-0.750hr=0.750hr.:***Nope,
3) average velocity vector=(-547km/h)i+(-1.31e3km/h)j***Nope
***Velocity=(-410km)/.75hr i+(-980km/1.5hr)j.***

(c) Average speed is merely a magnitude, so just add the magnitude of each trip and divide by the time interval.****Nope, add the two distances, then divide by total time. total time=2.25 hrs

2. An Earth satellite moves in a circular orbit 687 km above the Earth's surface. The period of the motion is 98.3 min.

(a) What is the speed of the satellite?
(b) What is the magnitude of the centripetal acceleration of the satellite?

(a) Using formula of T=2πr/v I solve for v. This yields v=2πr/T.
1) r=(mean earth radius)+(satellite height)=6,371,687m***NOPE NOPE
height=6.37E6+.687E6...

For question 2, it seems like you made a mistake in your calculation of the speed of the satellite.

(a) To find the speed of the satellite, you correctly used the formula T = 2πr/v, where T is the period, r is the radius of the orbit, and v is the speed of the satellite. Solving for v, you get v = 2πr/T.

1) r = (mean earth radius) + (satellite height) = 6,371,687 m + 687,000 m = 7,058,687 m
2) v = (2π * 7,058,687 m) / (98.3 min * 60 s/min) = 7,587.35 m/s (approximately)

So the correct speed of the satellite is approximately 7,587.35 m/s.

(b) Now, let's calculate the magnitude of the centripetal acceleration of the satellite. Centripetal acceleration is given by the formula a = v^2 / r, where a is the centripetal acceleration, v is the speed of the satellite, and r is the radius of the orbit.

1) a = (7,587.35 m/s)^2 / 7,058,687 m = 8.174 m/s^2 (approximately)

So, the magnitude of the centripetal acceleration of the satellite is approximately 8.174 m/s^2.

Please note that the values you provided are in reasonable ranges, so you may want to double-check your calculations to identify where the mistake occurred.