Starting from rest, a 2500 kg helicopter accelerates straight up at a constant 1.7 m/s2. What is the helicopter's height at the moment its blades are providing an upward force of 29 kN? The helicopter can be modeled as a 2.6-m-diameter sphere.
How do you start this question? Does it include gravity?
let g = 9.8 m/s^2
weight = 2500 * 9.8 = 24,500 N
Thrust - weight - drag = m a
29,000 - 24,500 - drag = 2500 (1.7)
drag = 250 N at this speed and thrust
Now I do not know what your class has done about drag of a sphere. However what is needed is the speed at which the drag of this 1.3 meter radius sphere is 250 Newtons in air.
then v = 1.7 t and h = (1/2)1.7 t^2
No, we have not covered Thrust. Thank you for your help.
To start solving this problem, we need to consider the relevant forces acting on the helicopter. The upward force provided by the helicopter's blades is balanced by the weight of the helicopter, which is due to gravity. Since the problem asks for the helicopter's height, gravity will be included in the calculations.
We can first calculate the weight of the helicopter by multiplying its mass (2500 kg) by the acceleration due to gravity (9.8 m/s^2).
Weight of helicopter = mass × acceleration due to gravity
= 2500 kg × 9.8 m/s^2
Next, we need to convert the upward force provided by the helicopter's blades from kilonewtons (kN) to newtons (N) by multiplying it by 1000.
Upward force provided by blades = 29 kN × 1000
= 29,000 N
Now, let's calculate the net force acting on the helicopter by subtracting the weight of the helicopter from the upward force provided by the blades.
Net force = upward force provided by blades - weight of helicopter
Finally, using Newton's second law (F = m × a), we can determine the acceleration of the helicopter by dividing the net force by its mass.
Acceleration = Net force / mass
Once we have the acceleration, we can use the kinematic equation for vertical motion:
Δy = v_0t + (1/2)at^2
Where:
Δy = change in vertical position (height)
v_0 = initial velocity (which is zero in this case since the helicopter starts from rest)
a = acceleration
t = time
By rearranging the equation, we can solve for the height (Δy) at the point when the helicopter's blades are providing an upward force of 29 kN.
To solve this question, we need to consider both the upward force provided by the helicopter's blades and the force of gravity acting on the helicopter.
First, let's calculate the weight of the helicopter using the formula:
Weight = mass × acceleration due to gravity
The acceleration due to gravity is approximately 9.8 m/s². Therefore, the weight of the helicopter is:
Weight = 2500 kg × 9.8 m/s² = 24500 N
Now, let's find the net force acting on the helicopter:
Net force = upward force - weight
Upward force = 29 kN = 29000 N
Net force = 29000 N - 24500 N = 4500 N
Since the helicopter is accelerating upward at a constant rate, we can use Newton's second law of motion:
Net force = mass × acceleration
Rearranging the equation to solve for mass:
Mass = Net force / acceleration
Mass = 4500 N / 1.7 m/s² ≈ 2647 kg
Now that we have the mass of the helicopter, we can calculate the height it reaches. To do this, we need to consider conservation of energy. The initial kinetic energy is zero since the helicopter starts from rest. The final kinetic energy is also zero when the helicopter reaches its maximum height, so it only has gravitational potential energy.
Gravitational potential energy is given by the formula:
Potential energy = mass × gravity × height
Setting the potential energy equal to the gained energy (work done by the upward force of the blades), we have:
Potential energy = Upward force × distance
Since the upward force acts over a distance equal to the height, we can rewrite the equation as:
Mass × gravity × height = Upward force × height
Rearranging the equation to solve for height:
Height = (Upward force × height) / (Mass × gravity)
Height = (29000 N × height) / (2647 kg × 9.8 m/s²)
Simplifying the equation, we can substitute the diameter to calculate the height:
Height = (29000 N × height) / (2647 kg × 9.8 m/s²) = (29000 N × (2.6 m / 2)) / (2647 kg × 9.8 m/s²)
Height ≈ 2.98 m
Therefore, the helicopter's height at the moment its blades are providing an upward force of 29 kN is approximately 2.98 meters.