Question

When a ball is dropped from a state of rest at time t=0t=0, the distance, measured in feet, that it has traveled after tt seconds is given by the formula s(t)=16t2s(t)=16t2 . (use decimal notation. if necessary, give your answer rounded to two decimal places.) (a) how far does the ball travel during the time interval [3, 3.5 ]? distance = .5 feet (b) compute the average velocity over the time interval [3, 3.5 ]. average velocity = 104 feet/sec (c) by computing the average velocity over the time intervals [3, 3.1 ], [3, 3.01 ], and [3, 3.001 ], . . . , estimate the ball's instantaneous velocity at t=3t=3. instantaneous velocity = feet/sec?

Answer 1

To calculate the distance and velocities for the ball in question, we use the formula s(t)=16t^2, calculate differences, and use limits as the intervals approach an instant.

Step-by-step explanation:

The formula s(t)=16t2 gives the distance s(t) in feet that a ball has traveled after t seconds when dropped from a state of rest. To find the distance during the time interval [3,3.5], we calculate s(3.5) and s(3), and then find the difference: s(3.5) = 16 × (3.5)2 and s(3) = 16 × (3)2.

The average velocity over the interval [3,3.5] is the change in distance divided by the change in time, which is computed as ∆ s/ ∆ t.

For the instantaneous velocity at t=3, we compute the average velocities over smaller and smaller intervals approaching the instant t=3. As these intervals become smaller, the average velocity approaches the instantaneous velocity.

Answer 2

Part A. To solve for the distance travelled during the interval, all we have to do is to plug in values of t = 3 and t = 3.5 in the equation and the difference would be the answer:

when t = 3: s = 16 (3)^2 = 144 m

when t = 3.5: s = 16 (3.5)^2 = 196 m

Therefore the distance travelled within the interval is:

196 m – 144 m = 52 m

Part B. The velocity is calculated by taking the 1st derivative of the equation. v = ds / dt

s = 16 t^2

ds / dt = 32 t = v

when t = 3: v = 32 (3) = 96 m / s

when t = 3.5: v = 32 (3.5) = 112 m / s

Therefore the average velocity is:

(96 + 112) /2 = 104 m / s

Part C. We can still use the formula v = 32 t and plug in the value of t = 3

v = 32 t = 32 (3)

v = 96 m / s