009A Sample Midterm 3, Problem 2

The position function  $s(t)=-4.9t^{2}+200$  gives the height (in meters) of an object that has fallen from a height of 200 meters.

The velocity at time  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=a}   seconds is given by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{t\rightarrow a} \frac{s(t)-s(a)}{t-a}}

(a) Find the velocity of the object when  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=3.}

(b) At what velocity will the object impact the ground?

Foundations:
1. What is the relationship between velocity  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)}   and position  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t)?}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)=s'(t)}
2. What is the position of the object when it hits the ground?
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t)=0}

Solution:

(a)

Step 1:
Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)}   be the velocity of the object at time  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t.}
Then, we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{v(3)} & = & \displaystyle{\lim_{t\rightarrow 3} \frac{s(t)-s(3)}{t-3}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9t^2+200-(-4.9(9)+200)}{t-3}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9t^2+44.1}{t-3}.} \end{array}}

Step 2:
Now, we factor the numerator to get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{v(3)} & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9t^2+44.1}{t-3}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9(t^2-9)}{t-3}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow 3} \frac{-4.9(t-3)(t+3)}{(t-3)}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow 3} -4.9(t+3)}\\ &&\\ & = & \displaystyle{6(-4.9) \text{ meters/second}.} \end{array}}

(b)

Step 1:
First, we need to find the time when the object hits the ground.
This corresponds to  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t)=0.}
This give us the equation
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -4.9t^2+200=0.}
When we solve for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t,}   we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2=\frac{200}{4.9}.}
Hence,   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=\pm \sqrt{\frac{200}{4.9}}.}
Since  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t}   represents time, it does not make sense for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t}   to be negative.
Therefore, the object hits the ground at  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=\sqrt{\frac{200}{4.9}}.}

Step 2:
Now, we calculate the velocity.
We have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{v\bigg(\sqrt{\frac{200}{4.9}}\bigg)} & = & \displaystyle{\lim_{t\rightarrow \sqrt{\frac{200}{4.9}}} \frac{s(t)-s\bigg(\sqrt{\frac{200}{4.9}}\bigg)}{t-\sqrt{\frac{200}{4.9}}}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow \sqrt{\frac{200}{4.9}}} \frac{-4.9t^2+200-(-4.9\big(\frac{200}{4.9}\big)+200)}{t-\sqrt{\frac{200}{4.9}}}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow \sqrt{\frac{200}{4.9}}} \frac{-4.9t^2+200}{t-\sqrt{\frac{200}{4.9}}}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow \sqrt{\frac{200}{4.9}}} \frac{-4.9\big(t-\sqrt{\frac{200}{4.9}}\big)\big(t+\sqrt{\frac{200}{4.9}}\big)}{t-\sqrt{\frac{200}{4.9}}}}\\ &&\\ & = & \displaystyle{\lim_{t\rightarrow \sqrt{\frac{200}{4.9}}} -4.9\bigg(t+\sqrt{\frac{200}{4.9}}\bigg)}\\ &&\\ & = & \displaystyle{-4.9\bigg(\sqrt{\frac{200}{4.9}}+\sqrt{\frac{200}{4.9}}\bigg)}\\ &&\\ & = & \displaystyle{-9.8\sqrt{\frac{200}{4.9}}\text{ meters/second}.} \end{array}}

Final Answer:
(a)     Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6(-4.9) \text{ meters/second}}
(b)     Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -9.8\sqrt{\frac{200}{4.9}}\text{ meters/second}}

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