# If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height in meters t seconds later is given by y= 10t - 1.86t^2.(a) Find the average velocity over the given time...

If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height in meters t seconds later is given by y= 10t - 1.86t^2.

(a) Find the average velocity over the given time intervals:

(i) [1, 2] (ii) [1, 1.5] (iii) [1, 1.1]

(iv) [1, 1.01] (v) [1, 1.001]

(b) Estimate the instantaneous velocity when t = 1.

When a rock is thrown upwards on the planet Mars with an initial velocity of 10 m/s, its height after t seconds is given by y = 10t - 1.86t^2.

To estimate the average velocity of the rock in any time interval [a, b], first determine the distance traveled by the rock in [a, b]. This is 10b - 1.86b^2 - (10a - 1.86a^2). The time taken for this is (b-a). The average velocity is `(10b - 1.86b^2 - 10a + 1.86a^2)/(b-a)`

(i) [1,2] The average velocity is `(10*2 - 1.86*2^2 - 10*1+ 1.86*1^2)/(2-1)`

= `4.42/1`

= 4.42. m/s

(ii) [1,1.5] The average velocity is `(10*1.5 - 1.86*1.5^2 - 10*1+ 1.86*1^2)/(1.5-1)`

= `2.675/0.5`

= 5.35 m/s

(iii) [1,1.1] The average velocity is `(10*1.1 - 1.86*1.1^2 - 10*1+ 1.86*1^2)/(1.1-1)`

= `0.6094/0.1`

= 6.094 m/s

(iv) [1,1.01] The average velocity is `(10*1.01 - 1.86*1.01^2 - 10*1+ 1.86*1^2)/(1.01-1)`

= `0.062614/0.01`

= 6.2614 m/s

(v) [1,1.001] The average velocity is `(10*1.001 - 1.86*1.001^2 - 10*1+ 1.86*1^2)/(1.001-1)`

= `(6.27814*10^-3)/0.001`

= 6.27814 m/s

At t = 1, the instantaneous velocity is `y'_(t=1) = (10 - 2*1.86*t)_(t = 1)`

= 6.28 m/s