was wondering if someone can help me out. Question: a jet has a length of 59.7 meters. The runway on which the plane lands intersects another runway. The width of the intersection is 25m. the plane decelerates through the intersection at a rate of 5.70 m/s^2 and clears it with a final speed of 45 m/s. how much time is needed for the lkplane to clear the intersection?
vf2 = vi2 + 2ad
(45m/s)2 = vi2 + 2(-5.7m/s^)(25m)
Solve for Vi
Find the average velocity = (vi+vf)/2
Divide the width of the intersection by vavg
Oops, I did not take the length of the plane into account. My first solution shrank the plane to a point concentrated at the end of its tail. I think this can be fixed by changing the distance, d, from 25m to (25+59.7) m.
To find out how much time is needed for the plane to clear the intersection, you can use the kinematic equation:
v_f = v_i + at
Where:
- v_f is the final velocity (45 m/s in this case)
- v_i is the initial velocity (the speed of the plane when it enters the intersection, which we don't know yet)
- a is the acceleration (deceleration in this case, -5.7 m/s^2)
- t is the time we want to find
First, let's find the initial velocity of the plane. We know that the plane decelerates through the intersection, so the initial velocity is higher than the final velocity. Therefore, we can rearrange the equation:
v_i = v_f - at
Substituting the known values:
v_i = 45 m/s - (-5.7 m/s^2) * t
Now, we need to find the time it takes for the plane to clear the intersection. We know that the length of the plane is 59.7 meters, and the width of the intersection is 25 meters. Therefore, the total distance the plane needs to travel to clear the intersection is:
d = 59.7 m + 25 m
To calculate the time needed, we can use the equation:
d = v_i * t + 0.5 * a * t^2
Rearranging the equation:
t^2 + (2 * v_i / a) * t - (2 * d / a) = 0
Now we have a quadratic equation. Plugging in the known values:
t^2 + (2 * (45 m/s - (-5.7 m/s^2) * t) / (-5.7 m/s^2)) * t - (2 * (59.7 m + 25 m) / (-5.7 m/s^2)) = 0
This equation can be solved using the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
Where a = 1, b = (2 * (45 m/s - (-5.7 m/s^2) * t) / (-5.7 m/s^2)), and c = (2 * (59.7 m + 25 m) / (-5.7 m/s^2)).
However, solving this quadratic equation would involve iteration or numerical methods due to the presence of the variable t on both sides. Hence, the exact time needed to clear the intersection may not have a simple analytical solution.
Alternatively, you can plot the function and find the intersections with the y-axis (time axis) using numerical methods or software, such as graphing calculators or spreadsheet programs.