Question

A ball is thrown into the air by a baby alien on a planet in the system of Alpha Centauri with a velocity of 20 ft/s. Its height in feet after t seconds is given by

y
=
20
t
−
16
t
2
.

A. Find the average velocity for the time period beginning when t=1 and lasting

.01 s:
ft/s
Hint: This first interval is
[
1
,
1
+
.01
]
.
.005 s:
ft/s
.002 s:
ft/s
.001 s:
ft/s
NOTE: For the above answers, you may have to enter 6 or 7 significant digits if you are using a calculator.

Estimate the instantaneous velocity when t=1.
ft/s

Answer 1

The average velocity for the time period = (2.508977e-03) / (0.001) = 2.508977 ft/s

To find the average velocity for the given time periods, we can use the formula:

average velocity = (change in position) / (change in time)

For the time period beginning when t = 1 and lasting 0.01 s:

y(1.01) - y(1) = (43(1.01) - 23 ) - (43(1) - 23 )

= -0.2020202

average velocity = (-0.2020202) / (0.01) = -20.20202 ft/s

For the time period beginning when t = 1 and lasting 0.005 s:

y(1.005) - y(1) = (43(1.005) - 23(1.005 ) - (43(1) - 23(1 )

= 0.020100

average velocity = (0.020100) / (0.005) = 4.02 ft/s

For the time period beginning when t = 1 and lasting 0.002 s:

y(1.002) - y(1) = (43(1.002) - 23(1.002) ) - (43(1) - 23(1 )

= 0.010020

average velocity = (0.010020) / (0.002) = 5.01 ft/s

For the time period beginning when t = 1 and lasting 0.001 s:

y(1,001) - y(1) = (43(1,001) - 23(1,001 2) - (43(1) - 23(1 )

= 2.508977e-03

average velocity = (2.508977e-03) / (0.001) = 2.508977 ft/s

The estimated instantaneous velocity when t = 1 is -3 ft/s.

To estimate the instantaneous velocity when t = 1, we can find the derivative of y with respect to t, and evaluate it at t = 1:

y(t) = 43t - 232

y'(t) = 43 - 46t

y'(1) = 43 - 46(1) = -3

Therefore, the estimated instantaneous velocity when t = 1 is -3 ft/s.