If a ball is thrown in the air with a velocity 52 ft/s, its height in feet t seconds later is given by y = 52t - 16t^2. Find the average velocity ofr the time period beginning when t = 2 and lasting 0.5 second. ft/s 0.1 second. ft/s 0.05 second. ft/s 0.01 second. ft/s Estimate the instantaneous velocity when t = 2. ft/s

a.) - 20ft/s

b.) - 13.6ft/s

c.) - 12.8ft/s

d.) - 12.16ft/s

e.) - 12ft/s

Step-by-step explanation:

Average Velocity = Change in distance/change in time.

Distance from the question in given in form of t as y = 52t - 16t² If our initial time is 2, distance travelled at t=2 is given as

Y (2) = 52 (2) - 16 (2) ² = 104 - 64

Y = 40ft.

For question a, when change In t is 0.5 seconds, that is from 2 sec to 2.5 seconds,

Average velocity = y (2.5) - y (2) / 0.5

y (2.5) = 52 (2.5) - 16 (2.5) ² = 130 - 100 = 30

Y (2) = 40

Average velocity in 0.5 seconds = [30 - 40]/0.5 = - 20ft/s.

For question b, when change In t is 0.1 seconds, that is from 2 sec to 2.1 seconds,

Average velocity = y (2.1) - y (2) / 0.1

y (2.1) = 52 (2.1) - 16 (2.1) ² = 109.2 - 70.56 = 38.64

Y (2) = 40,

Average velocity in 0.1 seconds = [38.64 - 40]/0.1 = - 13.6ft/s.

For question c, when change In t is 0.05 seconds, that is from 2 sec to 2.05 seconds,

Average velocity = y (2.05) - y (2) / 0.05

y (2.05) = 52 (2.05) - 16 (2.05) ² = 106.6 - 67.24 = 39.36

Y (2) = 40

Average velocity in 0.05 seconds = [39.36 - 40]/0.05 = - 12.8ft/s.

For question d, when change In t is 0.01 seconds, that is from 2 sec to 2.01 seconds,

Average velocity = y (2.01) - y (2) / 0.01

y (2.01) = 52 (2.01) - 16 (2.01) ² = 104.52 - 64.6416 = 39.8784

Y (2) = 40

Average velocity in 0.5 seconds = [39.8784 - 40]/0.01 = - 12.16ft/s.

Instantaneous velocity at t = 2 is derived by getting the first derivative of y and inserting Our value of t=2 into the first derivative.

If y = 52t - 16t², then derivative of y becomes y' given as

y' = 52 - 32t

At t = 2,

y' = 52 - 32 (2) = 52 - 64 = - 12ft/s.

Instantaneous velocity at t=2 is given as - 12ft/s.