A plane is found by radar to be flying 5.9 km above the ground. The angle of elevation from the radar to the plane is 78.6°. Ten seconds later, the plane is directly over the station. Find the speed of the plane, assuming that it is flying level.
To find the speed of the plane, we can use the following steps:
Step 1: Find the horizontal distance covered by the plane in 10 seconds.
Step 2: Use the horizontal distance and the time to find the speed of the plane.
Step 1: Find the horizontal distance covered by the plane in 10 seconds.
In the 10 seconds, the horizontal distance covered by the plane is equal to the distance between the radar station and the point directly below the plane.
We can use trigonometry to find this distance.
Let's call this distance "x".
Using the angle of elevation, we can set up the following equation:
tan(78.6°) = x / 5.9 km
To solve for x, we can multiply both sides of the equation by 5.9 km:
x = 5.9 km * tan(78.6°)
x ≈ 29.36 km
So, the horizontal distance covered by the plane in 10 seconds is approximately 29.36 km.
Step 2: Use the horizontal distance and the time to find the speed of the plane.
The speed of the plane can be calculated by dividing the horizontal distance by the time taken.
Given that the plane covers a horizontal distance of 29.36 km in 10 seconds, we can calculate its speed:
Speed = Distance / Time
Speed = 29.36 km / 10 s
Speed ≈ 2.936 km/s
Therefore, the speed of the plane, assuming it is flying level, is approximately 2.936 km/s.
To find the speed of the plane, we can use the information given about its vertical and horizontal motion.
First, let's consider the vertical motion. The plane is initially 5.9 km above the ground, and 10 seconds later it is directly over the radar station. This means that in 10 seconds, the plane has descended 5.9 km. We can use the formula for vertical motion to find the vertical velocity:
Vertical displacement = (initial vertical velocity) * (time) + (0.5) * (acceleration due to gravity) * (time^2)
Since the plane is flying level, its vertical velocity is the same throughout the motion, but it is facing downwards due to gravity. The acceleration due to gravity is approximately 9.8 m/s^2. Converting the units:
5.9 km = 5900 m
Substituting the known values into the equation:
5900 = (initial vertical velocity) * 10 + (0.5) * 9.8 * (10^2)
Simplifying the equation:
5900 = 10(initial vertical velocity) + 490
Rearranging the equation:
10(initial vertical velocity) = 5900 - 490
10(initial vertical velocity) = 5410
(initial vertical velocity) = 5410 / 10
(initial vertical velocity) = 541 m/s
Next, let's consider the horizontal motion. The plane is flying level, so it does not experience any acceleration horizontally. We can use the following formula to find the horizontal distance traveled by the plane:
Horizontal distance = (horizontal velocity) * (time)
The time is 10 seconds, and we want to find the horizontal velocity. Let's calculate the horizontal distance:
Horizontal distance = (horizontal velocity) * 10
Since the plane is directly over the radar station 10 seconds later, the horizontal distance is equal to the distance between the radar station and the plane. Assume this distance is x meters.
x = (horizontal velocity) * 10
Now, we can use trigonometry to relate the vertical and horizontal velocities of the plane. The angle of elevation from the radar to the plane is 78.6 degrees. The vertical velocity and the horizontal velocity are related by the tangent of this angle:
tan(78.6) = (initial vertical velocity) / (horizontal velocity)
Simplifying the equation:
tan(78.6) = 541 / (horizontal velocity)
Now, we can solve for the horizontal velocity:
(horizontal velocity) = 541 / tan(78.6)
(horizontal velocity) ≈ 134.96 m/s
Therefore, the speed of the plane, assuming that it is flying level, is approximately 134.96 m/s.
distance flown if level = d
tan 78.6 = 5900/d
d = 1190 meters in ten seconds
so
119 meters/second