An engineer is designing a runway for an airport. Several planes will use the runway and the
engineer must design it so that it is long enough for the largest planes to become airborne before the runway ends. The largest plane accelerates at 3.30 m/s2 and has a takeoff speed of 88.0 m/s.
a) Sketch the velocity vs. time graph for this problem and paste your sketch in the space below.
b) From your graph, what is the minimum allowed length for the runway?
To determine the minimum allowed length for the runway, we need to analyze the velocity versus time graph. Here's how to do it:
a) Sketching the velocity vs. time graph:
To sketch the graph, we need to understand the variables involved and the relationships between them. In this case, the relevant variables are time (t) and velocity (v), with the plane accelerating at a constant rate of 3.30 m/s^2.
Initially, at t = 0, the plane's velocity is 0 m/s. As time passes, the velocity increases linearly due to the acceleration. At some point, the velocity reaches the takeoff speed of 88.0 m/s. As soon as the plane reaches this velocity, it becomes airborne and leaves the runway.
The graph will have time (t) on the x-axis and velocity (v) on the y-axis. It will start at the origin (0,0) and form a straight line with a positive slope.
b) Determining the minimum allowed length for the runway:
To find the minimum allowed length for the runway, we need to determine how long it takes for the plane to reach its takeoff speed and become airborne. This can be done by calculating the time it takes for the velocity to reach 88.0 m/s.
At a constant acceleration of 3.30 m/s^2 and starting from rest (0 m/s), we can use the kinematic equation:
v = u + at,
where v is the final velocity, u is the initial velocity (0 m/s), a is the acceleration (3.30 m/s^2), and t is the time.
Rearranging the equation to solve for time:
t = (v - u) / a,
substituting the given values:
t = (88.0 m/s - 0 m/s) / 3.30 m/s^2,
t = 26.67 seconds (rounded to two decimal places).
Since we now know the time it takes for the plane to reach its takeoff speed, we can determine the minimum allowed length for the runway by multiplying the time by the takeoff speed:
minimum length = t * v,
minimum length = 26.67 seconds * 88.0 m/s,
minimum length = 2,357.76 meters (rounded to two decimal places).
Therefore, the minimum allowed length for the runway is approximately 2,357.76 meters.
The velocity time graph is a straight line passing through the origin with a slope of 3.3 m/s².
From the graph, it should be possible to read off time, t, it takes to reach 88 m/s.
From the equation of kinematics, calculate S, the distance travelled over the time period t:
u = initial speed = 0
a = acceleration = 3.3 m/s²
S = ut + (1/2)at²