Question

height of the helicopter
a police helicopter takes off from the roof of a building. after reaching its peak, it flies straight ahead at a constant height for a few minutes. then it quickly ascends to a higher level.
using three segments, sketch a possible graph that shows the height of the helicopter versus time.

Explanation:

Step1: Analyze the Helicopter's Motion

The helicopter's motion has three phases: take - off (ascending from the building's roof), flying at a constant height, and then ascending to a higher level.

• For the first segment (take - off from the building's roof), the height \(h\) increases with time \(t\). So the graph here is a line with a positive slope (since as \(t\) increases, \(h\) increases). Let's assume the building has a non - zero height \(h_0\), so at \(t = 0\), \(h=h_0\) (the height of the building's roof), and as \(t\) increases, \(h\) goes up until it reaches the peak height for the first part.
• The second segment is flying at a constant height. In this case, as time \(t\) increases, the height \(h\) remains the same. So the graph here is a horizontal line (slope \(= 0\)).
• The third segment is ascending to a higher level. Again, the height \(h\) increases with time \(t\), so the graph is a line with a positive slope (steeper or less steep depending on the rate of ascent, but since it ascends "quickly", the slope should be relatively large).

Step2: Sketch the Axes

• The x - axis represents time \(t\) (in minutes, for example) and the y - axis represents height \(h\) (in meters, for example). Label the x - axis as "Time (t)" and the y - axis as "Height (h)".

Step3: Draw the Three Segments

• First segment: Start at a point \((0,h_0)\) on the y - axis (where \(h_0\) is the height of the building's roof) and draw a line with a positive slope (going up from left to right) until the time \(t_1\) when it reaches the first peak height \(h_1\).
• Second segment: From the point \((t_1,h_1)\), draw a horizontal line (slope \( = 0\)) until time \(t_2\). This represents the time when it is flying at a constant height.
• Third segment: From the point \((t_2,h_1)\), draw a line with a positive slope (steeper than the first segment if it ascends quickly) until it reaches the new height \(h_2\) at time \(t_3\).

(Note: Since this is a sketching problem, the final answer is the description of the graph as above. If we were to represent it in a more visual way, we would have a graph with three parts: increasing, constant, increasing.)

Answer:

The graph has three parts:

1. A line with a positive slope starting from the height of the building's roof (non - zero y - intercept) as time increases until the first peak height.
2. A horizontal line (constant height) as time continues.
3. A line with a positive slope (relatively steep for quick ascent) as time increases to reach a higher height. The x - axis is time, the y - axis is height.