Question

a school shuttle increases its speed in a parking lot. then it moves at a constant speed for a while. then it slows down as it approaches a stop sign. using three segments, sketch a possible graph that shows the shuttles speed versus time.

Explanation:

Step1: Analyze the first segment (speeding up)

The shuttle increases speed over time. On a speed - time graph, an increasing speed with respect to time is represented by a line with a positive slope. Let's assume the time starts at \(t = 0\) and speed starts at \(v=0\) (or some initial non - zero speed, but starting from rest is a simple case). So, we draw a line that goes from the origin (or a starting point on the speed - axis) with a positive slope. For example, if we consider time on the x - axis and speed on the y - axis, from \(t = 0\) to \(t=t_1\), the line has a positive slope \(m_1>0\), so the equation of this segment can be approximated as \(v(t)=m_1t + v_0\) (if \(v_0 = 0\), then \(v(t)=m_1t\)).

Step2: Analyze the second segment (constant speed)

After speeding up, the shuttle moves at a constant speed. On a speed - time graph, a constant speed means that the speed does not change with time. So, this segment is a horizontal line (slope \(m_2 = 0\)). From \(t = t_1\) to \(t=t_2\), the speed \(v = v_1\) (where \(v_1\) is the speed reached at the end of the first segment), so the graph is a horizontal line \(v(t)=v_1\) for \(t_1\leq t\leq t_2\).

Step3: Analyze the third segment (slowing down)

As the shuttle approaches the stop sign, it slows down. A decreasing speed with respect to time is represented by a line with a negative slope on a speed - time graph. From \(t = t_2\) to \(t=t_3\) (where at \(t = t_3\), the speed \(v = 0\) as it stops), the line has a negative slope \(m_3<0\). The equation of this segment can be written as \(v(t)=m_3(t - t_2)+v_1\), and at \(t = t_3\), \(0=m_3(t_3 - t_2)+v_1\), so \(m_3=-\frac{v_1}{t_3 - t_2}\).

To sketch the graph:

• Draw the x - axis (labeled "Time") and y - axis (labeled "Speed").
• First segment: From the origin (or a starting point) draw a line going up to the right (positive slope) to represent the shuttle speeding up.
• Second segment: From the end - point of the first segment, draw a horizontal line to the right to represent the constant speed.
• Third segment: From the end - point of the second segment, draw a line going down to the right (negative slope) until it meets the time - axis (or reaches speed \(= 0\)) to represent the shuttle slowing down.

Answer:

The graph should have three segments: 1. A segment with a positive slope (speeding up), 2. A horizontal segment (constant speed), 3. A segment with a negative slope (slowing down until speed is 0). The actual sketch would look like a trapezoid - shaped graph (or a combination of a triangle - like first segment, rectangle - like second segment, and triangle - like third segment) on the speed - time coordinate system.