A meteor P is tracked by a radar observatory on the earth at O. When the meteor is directly overhead (theta = 90), the following observations are recorded: r = 80km, r'= -20km/s, and theta'= 0.4rad/s. (a) Determine the speed of the meteor and the angle, beta, which it's velocity vector makes with the horizontal. (b) Repeat with the same given quantities except that theta = 75.

Ok so I have no problem calculating the velocity. I got 37.7 km/s for both parts a and b because the angle changing wouldn't effect the magnitude of the velocity. My problem is that I can't figure out how to get beta.

(a) The speed is the vector sum of components r' (-20) and r*theta' (32), which is 37.7 km/s, as you have noted. The "angle with the horizontal" is the tangent of the angle that the velocity makes with r*theta' component, which is

tan^-1 (20/32) = 32.0 degrees

would beta be different for part b?

wait it would be different. wouldn't you just subtract 15 since it's starting from an angle with 15 less degrees?

To find the angle β, which is the angle between the velocity vector of the meteor and the horizontal, we'll use trigonometry.

Let's start with part (a) when θ = 90 degrees:
- Here, we know that r' is the radial velocity component (velocity along the line from the radar to the meteor).
- We're given r' = -20 km/s, which indicates that the meteor is moving away from the radar. The negative sign tells us that the meteor is getting farther from the radar instead of getting closer.
- We also know that r is the distance from the radar to the meteor, which is constant and given as 80 km.

To find β, we'll use the equation tan(β) = r' / r.
- tan(β) = (-20 km/s) / (80 km)
- Simplifying, tan(β) = -0.25

To find β, we take the arctan of both sides: β = arctan(-0.25)
- Using a calculator, we find β ≈ -14.04 degrees (rounded to two decimal places).

Note: The negative sign indicates that the velocity vector of the meteor is pointing downwards relative to the horizontal when it is directly overhead.

For part (b) when θ = 75 degrees:
- Everything remains the same, except θ is 75 degrees instead of 90.
- Now we need to consider the angle change in our calculations.

We'll use the same equation tan(β) = r' / r, but we'll incorporate the angular velocity, θ', into our calculation:
- tan(β) = (r' + r * θ') / r
- tan(β) = (-20 km/s + 80 km * 0.4 rad/s) / 80 km
- tan(β) = (-20 km/s + 32 km/s) / 80 km
- tan(β) = 12 km/s / 80 km
- Simplifying, tan(β) = 0.15

Again, we take the arctan of both sides: β = arctan(0.15)
- Using a calculator, we find β ≈ 8.53 degrees (rounded to two decimal places).

So, for part (a) β ≈ -14.04 degrees, and for part (b) β ≈ 8.53 degrees.