Question

1. the graph gives the position s(t) of an object moving along a line at time t, over a 2.5-second interval. find the average velocity of the object over the following intervals.

a. 0.5, 2.5
b. 0.5, 2
c. 0.5, 1.5
d. 0.5, 1
(graph: s(t) vs t, with t=0.5 (s=46), t=1 (s=84), t=1.5 (s=114), t=2 (s=136), t=2.5 (s=150))

Explanation:

Part a: Interval \([0.5, 2.5]\)

Step1: Recall average velocity formula

The average velocity over \([a, b]\) is \(\frac{s(b) - s(a)}{b - a}\). Here, \(a = 0.5\), \(b = 2.5\), \(s(0.5)=46\), \(s(2.5)=150\).

Step2: Substitute values into formula

\(\frac{150 - 46}{2.5 - 0.5}=\frac{104}{2}=52\)

Part b: Interval \([0.5, 2]\)

Step1: Identify values

\(a = 0.5\), \(b = 2\), \(s(0.5)=46\), \(s(2)=136\).

Step2: Apply average velocity formula

\(\frac{136 - 46}{2 - 0.5}=\frac{90}{1.5}=60\)

Part c: Interval \([0.5, 1.5]\)

Step1: Determine values

\(a = 0.5\), \(b = 1.5\), \(s(0.5)=46\), \(s(1.5)=114\).

Step2: Calculate average velocity

\(\frac{114 - 46}{1.5 - 0.5}=\frac{68}{1}=68\)

Part d: Interval \([0.5, 1]\)

Answer:

s:
a. \(\boldsymbol{52}\)
b. \(\boldsymbol{60}\)
c. \(\boldsymbol{68}\)
d. \(\boldsymbol{76}\)