Question

rank segments a, b, and c from least acceleration to greatest acceleration. least greatest

Explanation:

To determine the acceleration for each segment, we use the formula for acceleration from a velocity - time graph, which is the slope of the velocity - time line. The formula for slope (acceleration, \(a\)) between two points \((t_1,v_1)\) and \((t_2,v_2)\) is \(a=\frac{v_2 - v_1}{t_2 - t_1}\).

Step 1: Analyze Segment A

For segment A, the initial time \(t_1 = 0\ s\) and initial velocity \(v_1=0\ m/s\). The final time \(t_2 = 1\ s\) and final velocity \(v_2 = 15\ m/s\).
Using the acceleration formula \(a_A=\frac{v_2 - v_1}{t_2 - t_1}=\frac{15 - 0}{1 - 0}=15\ m/s^2\)

Step 2: Analyze Segment B

For segment B, the initial time \(t_1 = 1\ s\) and initial velocity \(v_1 = 15\ m/s\). The final time \(t_2=2\ s\) and final velocity \(v_2 = 20\ m/s\).
Using the acceleration formula \(a_B=\frac{v_2 - v_1}{t_2 - t_1}=\frac{20 - 15}{2 - 1}=5\ m/s^2\)

Step 3: Analyze Segment C

For segment C, the velocity is constant (the graph is a horizontal line). So, the change in velocity \(\Delta v=v_2 - v_1=20 - 20 = 0\ m/s\) and the change in time \(\Delta t=t_2 - t_1\) (any non - zero time interval). Then, using the acceleration formula \(a_C=\frac{\Delta v}{\Delta t}=\frac{0}{\Delta t}=0\ m/s^2\)

Step 4: Analyze Segment D (for reference, though not needed for ranking A, B, C)

For segment D, the initial time \(t_1 = 4\ s\) and initial velocity \(v_1 = 20\ m/s\). The final time \(t_2=5\ s\) and final velocity \(v_2 = 0\ m/s\).
Using the acceleration formula \(a_D=\frac{v_2 - v_1}{t_2 - t_1}=\frac{0 - 20}{5 - 4}=- 20\ m/s^2\) (the negative sign indicates deceleration, but we are interested in the magnitude for the ranking of acceleration magnitude? Wait, no, acceleration is a vector, but when we talk about the "amount" of acceleration in terms of the slope, for ranking from least to greatest, we consider the magnitude of the slope (or the value of acceleration, considering sign for direction, but in terms of the rate of change). But in the case of segment C, acceleration is 0. Segment B has an acceleration of \(5\ m/s^2\), segment A has an acceleration of \(15\ m/s^2\)

Now, ranking from least acceleration to greatest acceleration:
Least acceleration: C (acceleration \(= 0\ m/s^2\)), then B (acceleration \(= 5\ m/s^2\)), then A (acceleration \(= 15\ m/s^2\))

Answer:

Least: C, B, A (in the order of least to greatest, so the first box is C, second is B, third is A)
Greatest: A (the last box should be A)

So the ranking from least to greatest acceleration for segments A, B, C is C, B, A.