Question

for the graph above, answer the following questions:
a) which graph is moving the slowest. (diamonds, triangles, or rectangles)? explain your answer:
b) which graph(s) is(are) moving in the negative direction? explain your answer:
c) if the “race” were between triangle and diamond, who started in the lead?
d) what does the intersection of the diamond and the triangle lines mean happened at that point?
e) what is each object’s starting position? ____ triangle __ diamond ____ rectangle

Explanation:

Step1: Analyze speed (a)

Speed is determined by the slope of the position - time graph. The formula for slope \(m=\frac{\Delta y}{\Delta x}\) (where \(y\) is position and \(x\) is time). For a position - time graph, a smaller magnitude of slope means slower speed. Looking at the graph, the diamonds have the smallest slope (the line is the least steep among the three non - negative sloped lines).

Step2: Determine negative direction (b)

In a position - time graph, a negative slope indicates motion in the negative direction. The rectangle - shaped data points form a line with a negative slope.

Step3: Identify the leader at start (c)

At \(t = 0\), the position of the triangle - shaped data points is \(y=1\) and the position of the diamond - shaped data points is \(y = 3\). So the diamond - shaped graph (diamonds) started in the lead.

Step4: Interpret intersection (d)

In a position - time graph, when two lines intersect, it means that at that particular time, the two objects have the same position.

Step5: Find starting positions (e)

For the triangle - shaped graph (triangle), at \(t = 0\), \(y=1\). For the diamond - shaped graph (diamond), at \(t = 0\), \(y = 3\). For the rectangle - shaped graph (rectangle), at \(t = 0\), \(y=22\).

Answer:

a) Diamonds. Because they have the smallest slope in the positive - sloped lines.
b) Rectangles. Because their line has a negative slope.
c) Diamonds. Since at \(t = 0\), diamonds have a greater position value.
d) The two objects (diamond - shaped and triangle - shaped) are at the same position at that time.
e) Triangle: \(1\) meter; Diamond: \(3\) meters; Rectangle: \(22\) meters.