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)

When the shuttle increases its speed, the speed - time graph should be a line with a positive slope (since speed is increasing with time). Let's assume the initial speed is \(v_0 = 0\) (starts from rest in the parking lot, approximately). So, for the first part of the graph, as time \(t\) increases, speed \(s\) increases linearly. For example, if we consider time on the x - axis and speed on the y - axis, the equation of this line could be \(s=mt\) where \(m>0\) (slope is positive).

Step2: Analyze the second segment (constant speed)

When the shuttle moves at a constant speed, the speed does not change with time. So, the graph in this segment is a horizontal line (slope = 0). If the speed reached after the first segment is \(v_1\), then for a time interval \(\Delta t\), the speed remains \(v_1\), so the equation is \(s = v_1\) (a horizontal line parallel to the time axis).

Step3: Analyze the third segment (slowing down)

When the shuttle slows down as it approaches the stop sign, the speed - time graph should be a line with a negative slope (since speed is decreasing with time). The speed will decrease until it reaches 0 (at the stop sign). So, if the time at the start of this segment is \(t_2\) and the speed is \(v_1\), the equation of this line could be \(s=v_1 - nt\) where \(n > 0\) (slope is negative) and at some time \(t_3\), \(s = 0\).

To sketch the graph:

• First segment: Start at the origin \((0,0)\) (or a small initial speed close to 0) and draw a line with a positive slope going up to a point \((t_1,v_1)\) where \(t_1\) is the time when it stops accelerating.
• Second segment: From \((t_1,v_1)\) draw a horizontal line to a point \((t_2,v_1)\) where \(t_2>t_1\) (this is the time when it starts decelerating).
• Third segment: From \((t_2,v_1)\) draw a line with a negative slope going down to the point \((t_3,0)\) where \(t_3>t_2\) (this is when it stops at the stop sign).

Answer:

The graph has three segments: 1) A line with positive slope (speeding up), 2) A horizontal line (constant speed), 3) A line with negative slope (slowing down to 0 speed). (The actual sketch would follow the above - described steps with appropriate axes and points, but in text - based form, this is the description of the speed - time graph for the shuttle's motion.)