Solving for Takeoff Distance: Airplanes A & B

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In summary, you are given two airplanes, A and B, with the same acceleration but B needing twice the speed to take off. Airplane A requires a runway of 600m to become airborne, and you are asked to find the length of the runway for airplane B and the time it takes for B to travel the length of its runway in terms of T, the time it takes for A to travel its runway. To solve this, you can create a velocity-time graph for both planes, with A reaching a takeoff speed of v and B needing a takeoff speed of 2v. Using the formula for the area of a triangle, you can find the length of the runway for B. Then, knowing that both planes have
  • #1
penguins
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Homework Statement


Airplane A, starting from rest with constant acceleration, requires a runway 600m long to become airborne. Airplane B requires a takeoff speed twice as great as that of airplane A, but has the same acceleration, and both planes start from rest.



Homework Equations


a.) How long must the runway be, in meters, for airplane B?
b.) If airplane A takes time T to travel the length of it runway, how long (in terms of T) will airplane B take to travel the length of its runway?


The Attempt at a Solution


I tried multiplying the 600m runway by 2 to get the runway length for airplane B, but that answer is incorrect, I am not sure how to find the length of the runway with the information given in the problem.
 
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  • #2
Welcome to PF;
Sketch a velocity-time diagram for each aircraft.
Their accelerations are the same, but B needs twice the speed.
The area under the graph is the displacement.
 
  • #3
How do you make a velocity vs time graph when you don't have the velocity or the time in the information given?
 
  • #4
penguins said:
How do you make a velocity vs time graph when you don't have the velocity or the time in the information given?

You know that airplane A requires a speed of V in order to takeoff, therefore, airplane B must reach a speed of 2V before it can take off. You know that both aircraft have the same acceleration, and you know how far aircraft A must roll before reaching speed V.

If only there were formulas which related acceleration, velocity, and distance traveled! Where could a poor student find such information like this? In a textbook, perhaps? Maybe on the web?
 
  • #5
You have to make up some marks on the graph.
Lets say that plane A's takeoff speed is v, put a mark roughly half-way up your velicity axis, and label it v.

Plane B's takeoff velocity is twice that, so put a mark about twice as far up the velocity axis and mark it 2v.

Lets say that plane A take time T to reach it's takeoff speed ... so put a mark about half-way along the t axis and mark it T.

You know the initial velocity of plane A - so you can draw a line from there to point (T,v).
Now you have a triangle - you are told that the area under that triangle is 600m.
You know the formula for the area of a triangle.

You know that the acceleration of plane B is the same as for plane A ... you should be able to take it from there.

Or you could just look up some equations and plug and chug.
 

1. What is the purpose of solving for takeoff distance for airplanes A & B?

The purpose of solving for takeoff distance for airplanes A & B is to determine the minimum amount of runway needed for the respective airplanes to safely take off. This calculation takes into account factors such as the aircraft's weight, speed, and air density, and is crucial for ensuring a safe and efficient takeoff.

2. How is takeoff distance calculated for airplanes A & B?

Takeoff distance for airplanes A & B is calculated by considering the aircraft's weight, speed, air density, and engine thrust. This data is used to determine the aircraft's lift and drag forces, and the resulting acceleration and deceleration during the takeoff process. The takeoff distance is then calculated by adding the distance required to reach takeoff speed, the distance required to climb to a safe altitude, and a safety margin for any unexpected variables.

3. What are some factors that can affect takeoff distance for airplanes A & B?

Some factors that can affect takeoff distance for airplanes A & B include the aircraft's weight, speed, air density, wind conditions, runway surface and gradient, and engine performance. These factors can vary depending on the specific aircraft and environmental conditions, and can greatly impact the required takeoff distance.

4. Why is it important to accurately solve for takeoff distance for airplanes A & B?

Accurately solving for takeoff distance for airplanes A & B is important for ensuring a safe and efficient takeoff. If the calculated takeoff distance is incorrect, the aircraft may not have enough runway to safely take off, leading to potential accidents and injuries. It is also important for pilots to have an accurate understanding of their aircraft's capabilities in order to make informed decisions during takeoff.

5. How can takeoff distance be reduced for airplanes A & B?

Takeoff distance for airplanes A & B can be reduced by optimizing various factors, such as reducing the aircraft's weight, increasing engine thrust, and choosing a runway with favorable conditions. Additionally, proper maintenance and performance checks of the aircraft can also help to improve its takeoff performance and reduce the required takeoff distance.

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