If a rock is thrown upward on the planet Mars with a velocity 12 m/s, its height in meters t seconds later is given by y = 12t − 1.86t2

Question

(a) Find the average velocity (in m/s) over the given time intervals.

[1,2]

[1,1.5]

[1,1.1]

[1,1.01]

[1,1.001

(b) Use your answers from part (a) to estimate the instantaneous velocity of the rock (in m/s) when t = 1.

Answer 1

At t = 1, the height is 12(1) - 1.86(1²) = 10.14

At t = 2, the height is 12(2) - 1.86(2²) = 16.56

Average velocity = Δy/Δt = (16.56 - 10.14)/(2 - 1) = 6.42 m/s

At t = 1, the height is 12(1) - 1.86(1²) = 10.14

At t = 1.5, the height is 12(1.5) - 1.86(1.5²) = 13.815

Average velocity = Δy/Δt = (13.815 - 10.14)/(1.5 - 1) = 3.675/0.5 = 7.35 m/s

At t = 1, the height is 12(1) - 1.86(1²) = 10.14

At t = 1.1, the height is 12(1.1) - 1.86(1.1²) = 10.9494

Average velocity = Δy/Δt = (10.9494 - 10.14)/(1.1 - 1) = 0.8094/0.1 = 8.094 m/s

At t = 1, the height is 12(1) - 1.86(1²) = 10.14

At t = 1.01, the height is 12(1.01) - 1.86(1.01²) = 10.222614

Average velocity = Δy/Δt = (10.222614 - 10.14)/(1.01 - 1) = 0.082614/0.01 = 8.2614 m/s

At t = 1, the height is 12(1) - 1.86(1²) = 10.14

At t = 1.001, the height is 12(1.001) - 1.86(1.001²) = 10.14827814

Average velocity = Δy/Δt = (10.14827814 - 10.14)/(1.001 - 1) = 0.00827814/0.001 = 8.27814 m/s

So, a guess might be that the instantaneous velocity at t = 1 is 8.28 m/s