A 727 jet needs to attain a speed of 200 mph to take off. If it can accelerate from 0 to 200 mph in 30 seconds, how long must the runway be? (Assume constant acceleration)
To find the length of the runway needed for the 727 jet to take off, we can use the kinematic equation:
v = u + at
Where:
v = final velocity (200 mph)
u = initial velocity (0 mph)
a = acceleration (unknown)
t = time taken to reach final velocity (30 seconds)
Rearranging the equation, we get:
a = (v - u) / t
Substituting the given values, we have:
a = (200 mph - 0 mph) / 30 seconds
a = 200 mph / 30 seconds
To convert mph to miles per minute, we divide by 60:
a = (200 mph / 30 seconds) * (1 minute / 60 seconds)
a = 200 / (30 * 60) miles per minute
a = 200 / 1800 miles per minute
Simplifying, we get:
a = 1/9 miles per minute
Now, we can use another kinematic equation to find the distance (s) travelled during acceleration:
s = ut + 1/2 at^2
Where:
s = distance travelled
u = initial velocity
t = time taken to reach final velocity
a = acceleration
Since the initial velocity is 0 mph, the equation can be simplified to:
s = 1/2 at^2
Substituting the known values:
s = 1/2 * (1/9 miles per minute) * (30 seconds / 60 seconds)^2
s = 1/2 * 1/9 * (1/2)^2
s = 1/2 * 1/9 * 1/4
s = 1/72 miles
Therefore, the runway length must be at least 1/72 miles, or approximately 0.014 miles.
To solve this problem, we can use the formula for acceleration:
acceleration = change in velocity / change in time
In this case, the change in velocity is the final velocity of 200 mph minus the initial velocity of 0 mph. The change in time is given as 30 seconds. Let's substitute these values into the formula:
acceleration = (200 mph - 0 mph) / 30 s
Now, we need to convert the velocity from mph to m/s, and time from seconds to hours to maintain a consistent unit:
acceleration = [200 mph * (1.60934 km/h) * (1000 m/1 km)] / [(30 s) * (3600 s/1 h)]
Simplifying this expression, we get:
acceleration = (200 * 1.60934 * 1000) / (30 * 3600)
Now, let's calculate the value of acceleration:
acceleration ≈ 8.94 m/s²
Given that you assumed constant acceleration, we can use the following kinematic equation:
velocity² = initial velocity² + 2 * acceleration * distance
Since the initial velocity is 0, the equation simplifies to:
velocity² = 2 * acceleration * distance
Plugging in the values we have:
(200 mph * (1.60934 km/h) * (1000 m/1 km))² = 2 * 8.94 m/s² * distance
Simplifying this equation gives us:
(320,000)² = 17.88 * distance
Solving for distance:
distance = (320,000)² / 17.88
Now, let's calculate the distance:
distance ≈ 5,729,954.6 meters
Therefore, the runway needs to be approximately 5,729,954.6 meters long for the 727 jet to attain a speed of 200 mph in 30 seconds, assuming constant acceleration.
average speed during takeoff = 100 mph
so 100 mph for 30 seconds
100 m/hr *1 hr/3600 seconds *30 seconds
= .833 mles