Question

1. during which time interval did the object move the fastest? *

position vs. time
0 - 2 hours
2 - 4 hours
4 - 6 hours
6 - 8 hours

Explanation:

Step1: Recall the relationship between speed and the slope of a position - time graph

Speed \(v=\frac{\Delta x}{\Delta t}\), where \(\Delta x\) is the change in position and \(\Delta t\) is the change in time. On a position - time graph, the slope of the line segment \(m = \frac{y_2 - y_1}{x_2 - x_1}\) (where \(y\) represents position and \(x\) represents time) gives the speed. A steeper slope means a higher speed.

Step2: Calculate the slopes for each time interval

• For the \(0 - 2\) hours interval: Let \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(2,20)\). The slope \(m_1=\frac{20 - 0}{2-0}=10\) miles per hour.
• For the \(2 - 4\) hours interval: The position is constant (\(y = 20\) miles for \(x\) from \(2\) to \(4\) hours). The slope \(m_2=\frac{20 - 20}{4 - 2}=0\) miles per hour.
• For the \(4 - 6\) hours interval: Let \((x_1,y_1)=(4,20)\) and \((x_2,y_2)=(6,0)\). The slope \(m_3=\frac{0 - 20}{6 - 4}=- 10\) miles per hour (the magnitude is \(10\) miles per hour).
• For the \(6 - 8\) hours interval: Let \((x_1,y_1)=(6,0)\) and \((x_2,y_2)=(8,80)\). The slope \(m_4=\frac{80 - 0}{8 - 6}=40\) miles per hour.

Answer:

\(6 - 8\) hours