Velocity Question

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 41 ft/s. Its height in feet after t seconds is given by y = 41 t - 26t^2 A. Find the average velocity for the time period beginning when t=3 and lasting .



.01 s:

.005 s:

.002 s:

.001 s:

Estimate the instantaneous velocity when t = 3

Answer 1

Let's get the instantaneous velocity first, as the average velocities will approach that.

The height function h is given by: h(t) = 41t - 26t²

It's the derivative of the height function evaluated at t = 3.

The derivative is: 41 - 52 t. So 41 - 52(3) = 41 - 156 = - 115 ft per second

Now for the average velocity from t = 3 to t = 3.01

Average velocity is (h(3.01) - h(3))/.01

which is: [41(3.01) - 26(3.01)² - 41(3) + 26*3]/.01

= [41(.01) - 26(06 + .0001)]/.01 = 41 - 26(6 + .01) = 41 - 6*26 - .26 = 41 - 156 - .26 = -115.26

The others can be done similarly, each one should get closer to the exact value of -115.

Answer 2

y= 41t - 26t^2

y' = -26t

y'(3) = -26(3) = -78 feet per second = instantaneous velocity at time t=3 seconds

average velocity for time slightly later from t=3 to t= 3+ is slightly less than -78 ft/sec