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$$
\begin{array}{l}{12-42 . \text { The velocity of a car is plotted as shown. Determine }} \\ {\text { the total distance the car moves until it stops }(t=80 \mathrm{s}) \text { . }} \\ {\text { Construct the } a-t \text { graph. }}\end{array}
$$

$$
\left.\begin{array}{l}{t=80 s} \\ {graph  \Rightarrow a-t}\end{array}\right\}
$$

$$
\left. \Delta S=\int v d t\right\}\rightarrow \text {area under the curve}
$$

\(∴ S=\frac{(10 * 40)}{المربع}+\frac{\frac{1}{2}(80-40) * 10}{المثلث}={600 \mathrm{m}} \)

$$
a=\frac{d v}{d t} \Rightarrow 0 \leq t<40 s
$$

$$
a=\frac{d v}{d t}=0
$$

$$
40<t \leq 80s
$$

$$
\frac{v- 1{0}}{t-40}=\frac{0-10}{80-40}
$$

\(∴ V=\left(\frac{1}{4} t+20\right) m / s\)

$$
a=\frac{d v}{d t}=\frac{d\left(-\frac{1}{4} t+20\right)}{d t}
$$

$$
=-\frac{1}{4}=-0.25 \mathrm{m} / \mathrm{s}^{2}
$$

$$
\begin{array}{l}{12-40 . \text { The jet car is originally traveling at a velocity }} \\ {\text { of } 10 \mathrm{m} / \mathrm{s} \text { when it is subjected to the acceleration shown. }} \\ {\text { Determine the car's maximum velocity and the time } t^{\prime} \text { when }} \\ {\text { it stops. When } t=0, s=0 \text { . }}\end{array}
$$

$$
V=10  m/s
$$

$$
V_{ma x}=?
$$

@ $$
t=0 \rightarrow \begin{array}{l}{s=0} \\ {v_0=10 m / s}\end{array}
$$

$$
\frac{d v}{d t}=0
$$

$$
d v=a d t
$$

$$
0 \leq t<15 s
$$

$$
a=6 m / s^{2}
$$

@ $$
t=0 \rightarrow v_{0}=10 m/s
$$

$$
\int_{10}^{v} d v=\int_{0}^{t} a d t
$$

$$
\left.\begin{array}{l}{v{-10}=6 t} \\ { v=6 t+10} \\ {@ \  t=15s}\end{array}\right\} \begin{array}{l}{v=6 * 15+10} \\ {v=100 \mathrm{m} / \mathrm{s}}\end{array}
$$

$$
15<t \leq t^{\prime}
$$

$$
a=-4 m/s^2
$$

@ $$
t=15s \Rightarrow V= 100{m/s} 
$$

$$
\int_{100}^{v} d v= \int_{15}^{t}-4 d t
$$

$$
v-100=-\left.4+\right|_{15} ^{t}=-4 t-(-60)
$$

$$
V=-4 t+60+100
$$

$$
=-4 t+160
$$

\(∴ v=0 \quad @ \ t=t^{\prime} \)

$$
0=-4 t^{\prime}+160
$$

$$
{t'}=40 \mathrm{S}
$$

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