a jetliner traveling northward is landing with a speed of 80 m/s. Once the jet touches down, it has 800 m of runway in which to reduce its speed to 10 m/s. (a) Compute the average acceleration (magnitude and direction) of the plane during landing (take the direction of the plane's motion as positive). (b) How long a runway does the jetliner need to stop?
WEll, it is headed north, with a negative acceleration.
it takes d=800 to slow down.
Vf^2=Vi^2+2ad
a= (100-6400)/2*800=54/15 m/s^2 S
what is d if Vf=0?
0^2=6400+2(54/15)d
To solve this problem, we'll use the equations of motion.
(a) To compute the average acceleration, we can use the equation:
average acceleration = change in velocity / time taken
The change in velocity can be calculated as the final velocity minus the initial velocity. The initial velocity is 80 m/s (since the plane is traveling northwards) and the final velocity is 10 m/s.
Change in velocity = 10 m/s - 80 m/s = -70 m/s
The negative sign indicates that the velocity is decreasing (opposite to the initial direction of motion).
Now, we need to find the time taken to achieve this change in velocity. We can use the equation:
change in velocity = average acceleration × time taken
Rearranging the equation to solve for time taken:
time taken = change in velocity / average acceleration
Substituting the values:
time taken = (-70 m/s) / average acceleration
Now, let's determine the average acceleration by dividing the change in velocity by the time taken. The negative sign is ignored in this step, as we are interested in the magnitude of acceleration.
average acceleration = 70 m/s / time taken
Since we don't have the value of time taken yet, we can't calculate the exact magnitude of acceleration. However, we can proceed to part (b) of the question and come back to calculate the magnitude in the final step.
(b) To find out how long a runway the jetliner needs to stop, we'll use the equation of motion:
final velocity^2 = initial velocity^2 + 2 × acceleration × distance
Since the jetliner is stopping, the final velocity is 0 m/s, the initial velocity is 80 m/s, and the distance is 800 m.
Rearranging the equation to solve for acceleration:
acceleration = (final velocity^2 - initial velocity^2) / (2 × distance)
Substituting the values:
acceleration = (0^2 - 80^2) / (2 × 800)
Now, we can calculate the acceleration:
acceleration = -6400 / 1600 = -4 m/s^2
The negative sign indicates that the acceleration is in the opposite direction to the initial motion.
Now we can go back to part (a) and calculate the magnitude of the average acceleration. Using the equation:
average acceleration = 70 m/s / time taken
we substitute the value of acceleration from part (b):
-4 m/s^2 = 70 m/s / time taken
Cross-multiplying:
-4 m/s^2 × time taken = 70 m/s
Simplifying:
time taken = 70 m/s / -4 m/s^2
time taken ≈ -17.5 s
Since time cannot be negative in this scenario, the negative sign indicates that the magnitude of acceleration and the direction of motion were chosen incorrectly. The correct answer is:
(a) The average acceleration of the plane during landing is approximately 4 m/s^2, in the opposite direction of its initial motion.
(b) To find out how long a runway the jetliner needs to stop, we need to take the absolute value of the time:
time taken ≈ 17.5 s
Therefore, the jetliner needs approximately 17.5 seconds to stop.