Question

1. rank the following:

a. rank the graphs according to which show the greatest average velocity from the beginning to the end of the motion. (zero is greater than negative, and ties are possible.)
most pos. v 1____ 2__ 3__ 4__ 5__ 6____ most neg. v
explain your reasoning for your ranking:
b. rank the graphs according to which show the greatest average speed from the beginning to the end of the motion.
greatest 1____ 2__ 3__ 4__ 5__ 6____ least
explain your reasoning for your ranking:

Explanation:

Part a

Step1: Recall the formula for average velocity

The formula for average velocity is \(v_{avg}=\frac{\Delta x}{\Delta t}\), where \(\Delta x = x_{f}-x_{i}\) and \(\Delta t=t_{f} - t_{i}\). Assume \(t_{i} = 0\) and \(t_{f}=10\ s\) for all graphs.

Step2: Calculate average velocity for each graph

• Graph A: \(x_{i}=25\), \(x_{f}=25\), so \(v_{avg}=\frac{25 - 25}{10-0}=0\)
• Graph B: \(x_{i}=5\), \(x_{f}=5\), so \(v_{avg}=\frac{5 - 5}{10-0}=0\)
• Graph C: \(x_{i}=0\), \(x_{f}=10\), so \(v_{avg}=\frac{10 - 0}{10-0}=1\)
• Graph D: \(x_{i}=3\), \(x_{f}=10\), so \(v_{avg}=\frac{10 - 3}{10-0}=0.7\)
• Graph E: \(x_{i}=15\), \(x_{f}=0\), so \(v_{avg}=\frac{0 - 15}{10-0}=- 1.5\)
• Graph F: \(x_{i}=0\), \(x_{f}\approx6\) (at \(t = 10\ s\)), \(\Delta x\) for the first - part (\(0\) to \(3\ s\)): \(x\) goes from \(0\) to \(10\), \(\Delta x_1=10\), \(\Delta t_1 = 3\); for the second - part (\(3\) to \(10\ s\)): \(x\) goes from \(10\) to \(6\), \(\Delta x_2=6 - 10=-4\). The total \(\Delta x=\Delta x_1+\Delta x_2=10+( - 4)=6\), \(v_{avg}=\frac{6-0}{10 - 0}=0.6\)

Step3: Rank the average velocities

Since \(1>0.7>0.6>0=- 0\>-1.5\)

Part b

Step1: Recall the formula for average speed

The formula for average speed is \(s_{avg}=\frac{d}{t}\), where \(d\) is the total distance traveled and \(t\) is the total time. Assume \(t = 10\ s\) for all graphs.

• Graph A: \(d=0\) (no movement), \(s_{avg}=0\)
• Graph B: \(d = 0\) (no movement), \(s_{avg}=0\)
• Graph C: \(d=10\), \(s_{avg}=\frac{10}{10}=1\)
• Graph D: \(d=10 - 3=7\), \(s_{avg}=\frac{7}{10}=0.7\)
• Graph E: \(d=15\) (moves from \(x = 15\) to \(x = 0\)), \(s_{avg}=\frac{15}{10}=1.5\)
• Graph F: \(d=\vert10 - 0\vert+\vert6 - 10\vert=10 + 4=14\), \(s_{avg}=\frac{14}{10}=1.4\)

Step2: Rank the average speeds

Since \(1.5>1.4>1>0.7>0 = 0\)

Answer:

a. \(1 - C\), \(2 - D\), \(3 - F\), \(4 - A\), \(5 - B\), \(6 - E\)
b. \(1 - E\), \(2 - F\), \(3 - C\), \(4 - D\), \(5 - A\), \(6 - B\)