It is common to see birds of prey rising upward on thermals. The paths they take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume a bird completes a circle of radius 8.00m every 5.00s and rises vertically at a rate of 3.00 m/s .
Find the speed of the bird relative to the ground.
Find the magnitude of the bird's acceleration
Find the direction of the bird's acceleration.
Find the angle between the bird's velocity vector and the horizontal.
Can someone please help with the forumla's?
The vertical motion and the horizontal motion are decoupled. The vertical velocity is v = 3 m/s and the vertical acceleration is 0 m/s^2
In the horizontal plane the speed is u = 2 pi r/T = 2 pi (8/5) = 10 m/s
and the acceleration is centripetal = v^2/r = 12.6 m/s^2 toward the center of the spiral
So the speed relative to ground is:
s = sqrt(u^2+v^2)
The acceleraation is all inward in the horizontal plane and is 12.6 m/s^2
tan theta = v/u = 3/10
given R, T and Vy (vertical velocity component).
A)
find horizontal velocity component
Vx=2πR/T
this is the horizontal component, use the pythagorean theorem with the vertical velocity given to find net speed: Vx^2 (we just found)+Vy^2(given upward velocity of bird)= V^2 (dont forget to sqrt V)
B)A=V^2/R
C) It's 0.
D) using the Vx and Vy we already found, we need to use trig to find the angle between the adjacent and the hypotenuse, so we use tan(theta)=opposite/adjacent, in this case opposite being vertical Vy and the adjacent being horizontal Vx.
So arctan(Vy/Vx)= theta.
gig 'em
Sure! I can help you with the formulas.
1. Speed of the bird relative to the ground:
The speed of the bird relative to the ground can be found by calculating the magnitude of the velocity vector. In this case, since the bird is moving in a circle and rising vertically, the velocity vector can be calculated as the sum of the horizontal and vertical components. The horizontal component remains constant throughout the motion and is equal to the circumference of the circle divided by the time taken to complete one revolution (since the time taken for one revolution is given as 5.00 seconds). Therefore, the horizontal velocity component is:
v_horizontal = (2π * 8.00 m) / 5.00 s
The vertical component is the constant upward velocity given as 3.00 m/s.
To find the speed of the bird relative to the ground, we can calculate the magnitude of the velocity vector using the Pythagorean theorem:
v = √(v_horizontal² + v_vertical²)
2. Magnitude of the bird's acceleration:
The magnitude of the bird's acceleration can be calculated using the formula for centripetal acceleration. In uniform circular motion, the centripetal acceleration is given by the equation:
a_centripetal = (v²) / r
where v is the velocity and r is the radius of the circular path. In this case, the velocity remains constant throughout the motion and is equal to v_horizontal. Therefore, the magnitude of the bird's acceleration is:
a = (v_horizontal²) / r
3. Direction of the bird's acceleration:
The direction of the bird's acceleration is towards the center of the circular path, which is directed inward. This is because the bird is constantly changing its direction to remain in a circular path.
4. Angle between the bird's velocity vector and the horizontal:
The angle between the bird's velocity vector and the horizontal can be found using trigonometry. Since we already know the horizontal and vertical components of the velocity vector, we can use the inverse tangent function to determine the angle:
θ = arctan(v_vertical / v_horizontal)
To solve this problem, we'll need to use some basic physics formulas for circular motion and acceleration.
1. Finding the speed of the bird relative to the ground:
The bird's speed relative to the ground can be determined by finding the magnitude of its velocity vector. In this case, the bird moves in a circle of radius 8.00m every 5.00s.
The formula for speed (v) in uniform circular motion is given by:
v = 2πr / T
where r is the radius and T is the time period.
Plugging in the values, we get:
v = (2π * 8.00m) / 5.00s
Calculating that, we find the speed of the bird relative to the ground.
2. Finding the magnitude of the bird's acceleration:
In uniform circular motion, the magnitude of acceleration (a) can be determined using the formula:
a = v^2 / r
where v is the velocity and r is the radius.
Using the values we found in step 1, we can calculate the magnitude of the bird's acceleration.
3. Finding the direction of the bird's acceleration:
The direction of the bird's acceleration is always directed towards the center of the circle, which is perpendicular to the bird's velocity vector.
4. Finding the angle between the bird's velocity vector and the horizontal:
To find the angle between the bird's velocity vector and the horizontal, we can use trigonometry. We know that the vertical component of the bird's velocity is its upward velocity (3.00 m/s), and the horizontal component is the speed we found in step 1.
Using trigonometry, we can calculate the angle between the velocity vector and the horizontal.
Once we have the values for these quantities, we can solve the given problem.