If a rock is thrown upward on the planet Mars with a velocity of 10 m / s, its height in meters

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If a rock is thrown upward on the planet Mars with a...

If a rock is thrown upward on the planet Mars with a velocity of $10 \mathrm{m} / \mathrm{s},$ its height in meters t seconds later is given by $\mathrm{y}=10 \mathrm{t}-1.86 \mathrm{t}^{2}$ (a) Find the average velocity over the given time intervals: $$\begin{array}{ll}{\text { (i) }[1,2]} & {\text { (ii) }[1,1.5]} \\ {\text { (iv) }[1,1.01]} & {\text { (v) }[1,1.001] {\text { (iii) }[1,1.1] }}\end{array}$$ (b) Estimate the instantaneous velocity when $t=1$

If a rock is thrown upward on the planet Mars with a velocity of $10 \mathrm{m} / \mathrm{s},$ its height in meters t seconds later is given by $\mathrm{y}=10 \mathrm{t}-1.86 \mathrm{t}^{2}$
(a) Find the average velocity over the given time intervals:
$$\begin{array}{ll}{\text { (i) }[1,2]} & {\text { (ii) }[1,1.5]} \\ {\text { (iv) }[1,1.01]} & {\text { (v) }[1,1.001] {\text { (iii) }[1,1.1] }}\end{array}$$
(b) Estimate the instantaneous velocity when $t=1$

Key Concepts

Limit Process

The limit process is used to formally define the derivative and instantaneous velocity. It involves taking the average velocity over smaller and smaller time intervals and observing the behavior as the interval length approaches zero. This method fundamentally underlies the transition from average to instantaneous measures of change in calculus.

Average Velocity

Average velocity is defined as the ratio of the change in displacement to the change in time over a given time interval. It is calculated by taking the difference between the object's position values at the endpoints of the interval and dividing by the duration of the interval. This concept is fundamental in understanding motion in kinematics and serves as an approximation of the object's speed over a non-instantaneous period.

Instantaneous Velocity

Instantaneous velocity refers to the velocity of an object at a specific moment in time. Unlike average velocity, which considers an interval, instantaneous velocity gives the exact rate at which the position is changing at a particular point. It provides a more precise description of motion when the velocity is not constant.

Derivative

In calculus, the derivative of a function at a point represents the instantaneous rate of change of that function with respect to its variable. When applied to a position function, the derivative yields the instantaneous velocity of the moving object. This concept bridges the calculation of average rates (over intervals) and instantaneous rates (at points).

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