Question

A helicopter starts from rest at point A and travels along the straight-line path with a constant acceleration a. If the speed v = 39 m/s when the altitude of the helicopter is h = 31 m, determine the values of r˙, r¨, θ˙, and θ¨ as measured by the tracking device at O. At this instant, θ = 34°, and the distance d = 160 m. Neglect the small height of the tracking device above the ground.

Answer 1

To find the values of r˙, r¨, θ˙, and θ¨, we need to understand the kinematics of the helicopter's motion. Given the constant acceleration and known values of speed and altitude, we can use the equations of motion to find the values of r˙, r¨, θ˙, and θ¨.

Step-by-step explanation:

To find the values of r˙, r¨, θ˙, and θ¨, we need to understand the kinematics of the helicopter's motion. Let's start with the definitions:

1. r is the distance of the helicopter from point O
2. θ is the angle between the helicopter's path and the direction of the vertical axis at point O
3. θ˙ and θ¨ are the first and second derivatives of θ with respect to time, respectively
4. r˙ and r¨ are the first and second derivatives of r with respect to time, respectively

Given that the helicopter has a constant acceleration a and the speed v is 39 m/s when the altitude h is 31 m, we can use the equations of motion to find the values of r˙, r¨, θ˙, and θ¨:

1. First, find the time taken to reach the altitude h using the equation s = ut + 0.5at^2, where s is the displacement, u is the initial velocity, t is the time, and a is the acceleration.
2. Next, find the distance travelled horizontally using the equation v = u + at, where u is the initial velocity, v is the final velocity, t is the time, and a is the acceleration.
3. Then, use the trigonometric relationships between the distance d, altitude h, and angle θ to find the values of r, θ, and θ˙.
4. Finally, differentiate the equations in step 3 to find the values of r˙, r¨, θ˙, and θ¨.