A transport plane takes off from a level landing field with two gliders in tow, one behind the other. The mass of each glider is 700 , and the total resistance (air drag plus friction with the runway) on each may be assumed constant and equal to 2600 . The tension in the towrope between the transport plane and the first glider is not to exceed 12000 .
a.) find the minimum length of the runway, if a speed of 40m/s is needed to takeoff.
b.)tension between 2 gliders while they are accelerating for takeoff
To find the minimum length of the runway, we need to consider the forces acting on the transport plane and the gliders.
Let's denote the mass of each glider as m = 700 kg, the total resistance on each glider as R = 2600 N, and the speed needed for takeoff as v = 40 m/s. Also, let the tension in the towrope between the transport plane and the first glider be T1 and the tension between the two gliders be T2.
a.) To find the minimum length of the runway, we need to determine the net force acting on the transport plane and calculate the distance it needs to travel to reach the required takeoff speed.
The net force on the transport plane is given by the equation F_net = F_thrust - F_drag - F_friction = ma, where a is the acceleration.
The thrust force provided by the plane is zero since it is a towed glider. Therefore, the net force simplifies to F_net = -F_drag - F_friction = ma.
For each glider, the drag force is given by F_drag = R = 2600 N, and the frictional force is also given by F_friction = R = 2600 N.
Therefore, the net force can be written as F_net = -2R = -2 * 2600 N = -5200 N.
Since the mass of the transport plane is not mentioned, we can assume that its mass is much greater than the combined mass of the two gliders. Therefore, its acceleration can be assumed to be negligible compared to the gliders' acceleration.
We can now use the equation F_net = ma to find the net force acting on the gliders:
-5200 N = (2m) * a
Substituting the given values, we have:
-5200 N = (2 * 700 kg) * a
Simplifying, we find:
a = -5200 N / (1400 kg) ≈ -3.71 m/s²
Now, we can find the distance traveled by the transport plane using the equation:
v² = u² + 2as
where u represents the initial speed and s represents the distance.
Since the transport plane starts from rest (u = 0), the equation simplifies to:
s = v² / (2 * |a|)
Substituting the given values, we have:
s = (40 m/s)² / (2 * 3.71 m/s²) ≈ 216.5 m
Therefore, the minimum length of the runway is approximately 216.5 meters.
b.) To find the tension between the two gliders while they are accelerating for takeoff, we can consider the net force acting on the second glider.
The net force on the second glider is given by the equation F_net = T2 - R - m * a, where T2 is the tension between the two gliders.
Substituting the given values, we have:
T2 - 2600 N - (700 kg) * (-3.71 m/s²) = 0
Simplifying, we find:
T2 = 2600 N + (700 kg) * (3.71 m/s²)
T2 ≈ 5347 N
Therefore, the tension between the two gliders while they are accelerating for takeoff is approximately 5347 N.
To solve this problem, we'll use Newton's second law of motion, which states that the sum of the forces acting on an object is equal to the mass of the object multiplied by its acceleration. In this case, we'll consider the forces acting on the gliders.
a.) To find the minimum length of the runway, we need to determine the acceleration of the gliders during takeoff. We can start by calculating the net force acting on the gliders.
The net force acting on each glider can be calculated using the equation:
Net Force = Thrust Force - Resistance Force
The Thrust Force is the tension in the towrope, which we can set equal to T.
Now, the Resistance Force can be calculated using the equation:
Resistance Force = Air Drag + Friction Force
Given that the total resistance force on each glider is constant and equal to 2600 N, we can rewrite the equation as:
Resistance Force = 2600 N
Since the force of air drag and the force of friction are equal, we can split the total resistance force evenly between them:
Air Drag = Friction Force = 2600 N / 2 = 1300 N
Now we can rewrite the equation for net force as:
Net Force = T - 1300 N
According to Newton's second law, we know that:
Net Force = Mass x Acceleration
For each glider, the mass is given as 700 kg. Therefore, we can rewrite the equation as:
T - 1300 N = 700 kg x Acceleration
The maximum tension in the towrope is given as 12000 N. So, we can set up an inequality:
T ≤ 12000 N
Now we have two equations:
T - 1300 N = 700 kg x Acceleration [Equation 1]
T ≤ 12000 N [Equation 2]
We can use these equations to find the acceleration and then calculate the minimum length of the runway using the formula:
Distance = (Initial Velocity^2) / (2 x Acceleration)
b.) To find the tension between the two gliders while they are accelerating for takeoff, we need to consider the forces acting on the second glider.
The net force acting on the second glider can be calculated using the equation:
Net Force = Tension - Resistance Force
We already know that the Resistance Force is 1300 N. Now we can rewrite the equation:
Net Force = Tension - 1300 N
Since the acceleration of the second glider is the same as the first glider (as both are being towed together), we can use Equation 1 from Part a to write:
Tension - 1300 N = 700 kg x Acceleration
By solving this equation, we can find the tension between the two gliders.