Question

If a ball is thrown into the air with a velocity of 34 ft/s, its height in feet t seconds later is given by y = 34t − 16t2.(a) Find the average velocity for the time period beginning when t = 2 and lasting for each of the following.(i) 0.5 seconds ft/s(ii) 0.1 seconds ft/s(iii) 0.05 seconds ft/s(iv) 0.01 seconds ft/s(b) Estimate the instantaneous velocity when t = 2. ft/s

Answer 1

To find the average velocity over different time intervals starting at t = 2 seconds, we apply the formula for average velocity. The instantaneous velocity at any time t can be obtained by differentiating the height function. For t = 2, the instantaneous velocity is -30 ft/s.

Step-by-step explanation:

Average and Instantaneous Velocity of a Thrown Ball

To find the average velocity over a time period, we use the formula for average velocity:

V

avg

= (y(t + Δt) - y(t)) / Δt

For the initial time t = 2 seconds and the different time intervals given, we can calculate:

1. For Δt = 0.5 seconds: Vavg = (y(2.5) - y(2)) / 0.5
2. For Δt = 0.1 seconds: Vavg = (y(2.1) - y(2)) / 0.1
3. For Δt = 0.05 seconds: Vavg = (y(2.05) - y(2)) / 0.05
4. For Δt = 0.01 seconds: Vavg = (y(2.01) - y(2)) / 0.01

The instantaneous velocity at any time t can be found by taking the derivative of the position function with respect to time:

V(t) = dy/dt = 34 - 32t

Thus, the instantaneous velocity at t = 2 is V(2) = 34 - 32(2) = -30 ft/s.

Answer 2

(a). (i)
`-38\ (ft)/(s)`; (ii)
`-31.6\ (ft)/(s)`; (iii)
`-30.8\ (ft)/(s)`; (iv)
`-30.16\ (ft)/(s)`

(b).
`-30\ (ft)/(s)`

Step-by-step explanation:

The formula you need to use is:

`V_(avg)=(y_2-y_1)/(t_2-t_1)`

Then, in this case:

`V_(avg)=((34t_2 -16(t_2)^2)-(34t_1 - 16(t_1)^2))/(t_2-t_1)`

(a) Susbtituting values, we get:

(i)
`V_(avg)=((34(2.5) - 16(2.5)^2)-(34(2)- 16(2)^2))/(2.5-2))=-38\ (ft)/(s)`

(ii)
`V_(avg)=((34(2.1) - 16(2.1)^2)-(34(2)- 16(2)^2))/(2.5-2))=-31.6\ (ft)/(s)`

(iii)
`V_(avg)=((34(2.05) - 16(2.05)^2)-(34(2)- 16(2)^2))/(2.5-2))=-30.8\ (ft)/(s)`

(iv)
`V_(avg)=((34(2.01) - 16(2.01)^2)-(34(2)- 16(2)^2))/(2.5-2))=-30.16\ (ft)/(s)`

(b) Observe that as the time after 2 second gets smaller, the average velocity gets closer to
`-30\ (ft)/(s)`. Therefpre, we can can estimate that the instantaneous velocity when
`t = 2` is:

`V=-30\ (ft)/(s)`