An aircraft has a take-off speed 180km/h and has length 250m from the start just before take-off a serious snag is detected in the engine and the aircraft failed to take-off. There is a wall at a distance 'd' = 230m ahead of the take-off point. The pilot applies breaks which produces as much retardation as the acceleration of the aircraft and succeeds in avoiding the crash on the wall. What is the reaction time of the pilot? Please tell me how to solve it

To solve this problem, we need to find the reaction time of the pilot, which is the time it takes for the pilot to apply the brakes after detecting the snag.

Let's break down the problem step by step:

1. Convert the take-off speed from km/h to m/s:
Take-off speed = 180 km/h = (180 * 1000) / 3600 m/s = 50 m/s.

2. Determine the initial velocity of the aircraft when the brakes are applied:
Since the aircraft failed to take off, the initial velocity is equal to the take-off speed, which is 50 m/s.

3. Determine the final velocity of the aircraft:
The final velocity is ultimately 0 m/s because the aircraft stops without crashing into the wall.

4. Find the average retardation or deceleration produced by the brakes:
The deceleration provided by the brakes is equal to the acceleration of the aircraft.

5. Calculate the distance covered by the aircraft during the reaction time:
The distance covered by the aircraft during the reaction time can be calculated using the equation:
Distance = Initial Velocity * Time + 0.5 * Deceleration * Time^2.

Since we are looking for the reaction time, we can use this equation to find it.

6. Plug in the given values into the equation and solve for time:
Distance = 250 m (given length)
Deceleration = Acceleration = (negative) Retardation = - Retardation.

Substitute these known values into the equation Distance = Initial Velocity * Time + 0.5 * Deceleration * Time^2:
250 = 50 * Time + 0.5 * Retardation * Time^2.

Rearranging the equation, you get:
0.5 * Retardation * Time^2 + 50 * Time - 250 = 0.

This is a quadratic equation that can be solved using either factoring, completing the square, or the quadratic formula.

Once you have solved the quadratic equation, you will have two possible values for time (since it's quadratic). Since we are interested in the reaction time, we select the positive value.

7. The positive value you obtain will be the reaction time of the pilot.

Now, you can follow the steps above to solve the problem and find the reaction time of the pilot.