Question

1. the table below shows the position of a motorcyclist after accelerating from rest.

t (seconds)	0	1	2	3	4	5	6
s (feet)	0	4.9	20.6	46.5	79.2	124.8	176.7

a. find the average velocity for each time period:
i. 2, 4
ii. 3, 4
iii. 4, 5
iv. 4, 6

Explanation:

Step1: Recall average - velocity formula

The average velocity $v_{avg}$ over the time interval $[t_1,t_2]$ is given by $v_{avg}=\frac{s(t_2)-s(t_1)}{t_2 - t_1}$, where $s(t)$ is the position function.

Step2: Calculate average velocity for $[2,4]$

We have $t_1 = 2$, $t_2=4$, $s(2)=20.6$, and $s(4)=79.2$.
$v_{avg}=\frac{s(4)-s(2)}{4 - 2}=\frac{79.2-20.6}{2}=\frac{58.6}{2}=29.3$ feet per second.

Step3: Calculate average velocity for $[3,4]$

We have $t_1 = 3$, $t_2 = 4$, $s(3)=46.5$, and $s(4)=79.2$.
$v_{avg}=\frac{s(4)-s(3)}{4 - 3}=\frac{79.2 - 46.5}{1}=32.7$ feet per second.

Step4: Calculate average velocity for $[4,5]$

We have $t_1 = 4$, $t_2 = 5$, $s(4)=79.2$, and $s(5)=124.8$.
$v_{avg}=\frac{s(5)-s(4)}{5 - 4}=\frac{124.8-79.2}{1}=45.6$ feet per second.

Step5: Calculate average velocity for $[4,6]$

We have $t_1 = 4$, $t_2 = 6$, $s(4)=79.2$, and $s(6)=176.7$.
$v_{avg}=\frac{s(6)-s(4)}{6 - 4}=\frac{176.7-79.2}{2}=\frac{97.5}{2}=48.75$ feet per second.

Answer:

i. 29.3 feet per second
ii. 32.7 feet per second
iii. 45.6 feet per second
iv. 48.75 feet per second