The distance (in feet) of an object from a point is given by s(t)=3t^(2)−5, where t is in seconds. Round your answer to two decimal places.
1. What is the average velocity of the object between t=1 and t=8.5?
2. By using smaller and smaller intervals around 1, estimate the instantaneous velocity at time t=1.
#1. That would be (s(8.5) - s(1))/(8.5 - 1)
#2. You can start with ∆t = 0.1
(s(1.1) - s(1)) / (1.1 - 1)
and decrease the ∆t
Thank you so much!! Great help!!
To find the average velocity between two points, we need to determine the change in position divided by the change in time.
1. Average velocity between t=1 and t=8.5:
To calculate the average velocity, we first need to find the position at t=1 and t=8.5 and then find the change in position and the change in time.
At t=1:
s(1) = 3(1)^2 - 5 = 3 - 5 = -2 feet
At t=8.5:
s(8.5) = 3(8.5)^2 - 5 = 3 * 72.25 - 5 = 216.75 - 5 = 211.75 feet
Change in position = s(8.5) - s(1) = 211.75 - (-2) = 213.75 feet
Change in time = 8.5 - 1 = 7.5 seconds
Average velocity = change in position / change in time = 213.75 / 7.5 ≈ 28.50 feet/second
Therefore, the average velocity of the object between t=1 and t=8.5 is approximately 28.50 feet/second.
2. Estimating instantaneous velocity at t=1:
To estimate the instantaneous velocity at t=1, we can use smaller intervals around t=1 to calculate the average velocity and see how it converges to the instantaneous velocity.
Let's calculate the average velocity using smaller intervals:
Interval 1: t=1 to t=1.5
Change in position = s(1.5) - s(1) = [3(1.5)^2 - 5] - [3(1)^2 - 5] = 11.25 - (-2) = 13.25 feet
Change in time = 1.5 - 1 = 0.5 seconds
Average velocity = change in position / change in time = 13.25 / 0.5 = 26.5 feet/second
Interval 2: t=1 to t=1.1
Change in position = s(1.1) - s(1) = [3(1.1)^2 - 5] - [3(1)^2 - 5] = 9.63 - (-2) = 11.63 feet
Change in time = 1.1 - 1 = 0.1 seconds
Average velocity = change in position / change in time = 11.63 / 0.1 = 116.3 feet/second
By using smaller and smaller intervals, we can see that the average velocity becomes closer to the instantaneous velocity.
Therefore, by estimating the average velocities over increasingly smaller intervals, we can get a better estimate of the instantaneous velocity at t=1.