A jet plane lands with a speed of 81 m/s and can accelerate at a maximum rate of −5.00 m/s2 as it comes to rest.
(a) From the instant the plane touches the runway, what is the minimum time needed before it can come to rest?
(b) Can this plane land on a small tropical island airport where the runway is 0.800 km long?
v = 0 = 81 - 5 t
800 = 81 t - 5.2 t^2
use that same stopping time t
To find the minimum time needed for the jet plane to come to rest, we can use the equations of motion.
(a) The equation that relates velocity, acceleration, and time is:
v = u + at,
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken. In this case, the final velocity is 0, the initial velocity is 81 m/s, and the acceleration is −5.00 m/s².
Plugging these values into the equation, we have:
0 = 81 + (-5)t.
Simplifying the equation, we get:
5t = 81.
Dividing both sides by 5, we find:
t = 16.2 s.
Therefore, the minimum time needed for the jet plane to come to rest is 16.2 seconds.
(b) To determine if the plane can land on a 0.800 km long runway, we need to find the distance it travels before coming to rest.
The equation that relates distance, initial velocity, acceleration, and time is:
s = ut + 0.5at²,
where s is the distance, u is the initial velocity, a is the acceleration, and t is the time taken.
Plugging in the values we have, the equation becomes:
s = 81t + 0.5(-5)t².
Since we know that the final velocity is 0, the distance traveled will be equal to the length of the runway. Converting 0.800 km to meters, we get:
s = 0.800 km * 1000 m/km = 800 m.
Now we can solve the equation:
800 = 81t + 0.5(-5)t².
Simplifying and rearranging, we have:
0.5(-5)t² + 81t - 800 = 0.
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b² - 4ac)) / 2a,
where a, b, and c are the coefficients in the equation. In this case, they are:
a = 0.5(-5) = -2.5,
b = 81,
c = -800.
Plugging these values into the quadratic formula, we get two possible values for t:
t = (-81 ± √(81² - 4(-2.5)(-800))) / 2(-2.5).
Evaluating this equation, we find:
t ≈ 3.36 s, t ≈ 23.9 s.
Since the time taken to come to rest is greater than 23.9 seconds, the plane cannot land on the 0.800 km long runway.
Therefore, the plane cannot land on the small tropical island airport where the runway is 0.800 km long.