Question

An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If trepresents the time in minutes since the plane has left the terminal, let \(x\left( t \right)\)be the horizontal distance traveled and \(y\left( t \right)\) be the altitude of the plane.

(a) Sketch a possible graph of\(x\left( t \right)\).

(b) Sketch a possible graph of\(y\left( t \right)\).

(c) Sketch a possible graph of the ground speed.

(d) Sketch a possible graph of the vertical velocity.

Short answer

(a) The graph of \(x\left( t \right)\)is as follows.

(b) The graph of \(y\left( t \right)\) is as follows

(c) The graph of ground speed is as follows.

(d) The graph of vertical velocity is as follows.

Step-by-step solution

(a) Step 1: Describe the given information

The time in minutes since the plane has left the terminal is\(t\)\(x\left( t \right)\)is the horizontal distance and\(y\left( t \right)\)is the altitude of the plane. It is required to sketch the graph of\(x\left( t \right)\).

Sketch the possible graph of\(x\left( t \right)\).

On the graph, \(x\left( t \right)\) will be on the vertical axis, and \(t\) will be on the horizontal axis.

Over the course of the flight, the distance traveled would continually increase from 0 to 400 miles. For this condition, the graph will be uphill from left to right.

After that, the graph will be steepest in the middle because the plane is traveling fastest. This means that the distance traveled is increasing fastest during the middle of the flight.

Draw the required graph of \(x\left( t \right)\)as follows.

(b) Step 3: Sketch the possible graph of\(y\left( t \right)\).

On the graph, \(y\left( t \right)\) will be on the vertical axis, and \(t\) will be on the horizontal axis.

At the beginning of the flight, the altitude would increase from 0 to approximately\(35,000\;{\rm{ft}}\). For this condition, the graph will be going uphill from left to right.

Next, the plane will be leveled off at cruising altitude. So, the graph will remain relatively flat for some time.

After that, the plane will descend to land at the end of 60 minutes. So, the graph will go downhill from \(35,000\;{\rm{ft}}\) to \(0\;{\rm{ft}}\) as the plane is landing.

Draw the required graph of \(x\left( t \right)\) as follows.

(c) Step 4: Sketch a possible graph of the ground speed

The ground speed is measured by taking a reference point on the ground.

It is given that the plane travels \(400\) miles per hour, so its average ground speed is \(400\) miles per hour.

However, throughout the flight, the plane would start from \(0\;{\rm{mph}}\) and increase to its top speed, then slow down as it comes in for a landing.

On the graph of the ground speed, let \(y\) be the speed, and \(t\) be the time throughout the flight.

The graph will be of the increasing pattern until reaching the highest ground speed, and then it will remain flat for some time as the plane flies steadily at its maximum speed, then the graph would go downhill as the plane's speed lessens while it comes in for a landing.

Draw the graph as follows.

(d) Step 5: Sketch a possible graph of the vertical velocity.

The vertical velocity is the measurement of speed and direction as the plane rises and falls. When the plane is ascending, the vertical velocity will be positive, and when the plane is descending, the vertical velocity will be negative.

Initially, the plane starts on the ground with a velocity of\(0\;{\rm{mph}}\). As the plane ascends, it increases its vertical velocity because it is climbing at a fast rate. The graph would show this change in vertical velocity by going uphill from left to right during this interval.

Eventually, the plane would approach its cruising altitude so its vertical velocity would decrease. The graph shows this decrease by going downhill. When it is cruising, the plane's height does not change much, so its vertical velocity would be around\(0\). For this interval of time, the graph would be flat.

After some time, the plane will start to descend. At this time there will be a negative vertical velocity. As the plane descends faster, the vertical velocity will decrease faster and becomes more negative. The graph shows this by going downhill from 0 into the negative.

As the plane comes in for a landing, it will start to descend less rapidly. Hence its vertical velocity will become less negative until the plane lands. The graph will show this by gradually going uphill and approaching 0. The vertical change is 0 when the plane land. So the graph will end on the horizontal axis.

Draw the graph as follows.