Questions: The position of an object moving vertically along a line is given by the function s(t)=-4.9t^2+30t+25. Find the average velocity of the object over the following intervals. a. [0,6] b. [0,5] c. [0,4] d. [0, h], where h>0 is a real number

The position of an object moving vertically along a line is given by the function s(t)=-4.9t^2+30t+25. Find the average velocity of the object over the following intervals.
a. [0,6]
b. [0,5]
c. [0,4]
d. [0, h], where h>0 is a real number

Transcript text: The position of an object moving vertically along a line is given by the function $\mathrm{s}(\mathrm{t})=-4.9 \mathrm{t}^{2}+30 \mathrm{t}+25$. Find the average velocity of the object over the following intervals. a. $[0,6]$ b. $[0,5]$ c. $[0,4]$ d. $[0, \mathrm{~h}]$, where $\mathrm{h}>0$ is a real number

Solution

Solution Steps

Step 1: Define the Position Function

The position of the object moving vertically along a line is given by the function: \[ s(t) = -4.9 t^2 + 30 t + 25 \]

Step 2: Calculate Average Velocity over the Interval $[0, 6]$

To find the average velocity over the interval $[0, 6]$, we compute: \[ \text{Average Velocity} = \frac{s(6) - s(0)}{6 - 0} \] The result is: \[ \text{Average Velocity over } [0, 6] = 0.6 \]

Step 3: Calculate Average Velocity over the Interval $[0, 5]$

Next, we calculate the average velocity over the interval $[0, 5]$: \[ \text{Average Velocity} = \frac{s(5) - s(0)}{5 - 0} \] The result is: \[ \text{Average Velocity over } [0, 5] = 5.5 \]

Step 4: Calculate Average Velocity over the Interval $[0, 4]$

Finally, we compute the average velocity over the interval $[0, 4]$: \[ \text{Average Velocity} = \frac{s(4) - s(0)}{4 - 0} \] The result is: \[ \text{Average Velocity over } [0, 4] = 10.4 \]