10. Physics Applications of Integrals

At $t=0$, a car approaching a stop sign decelerates from a speed of 50 $\mathrm{mi}$/$hr$ according to the acceleration function $a\left(t\right)=4t+3$, where $t\ge 0$ and is measured in hours. How far does the car travel between $t=0$ and $t=0.1$ $hr$?

Suppose that a particle travels along the $x$-axis and its velocity is given by $v\(\left\)(t\(\right\))=2\(\cos\) t$ for $0\(\le\) t\(\le\)2\(\pi\)$.

Find total distance traveled by the particle on $\(\left\]\lbrack\)0,\(\frac{5\pi}{4}\[\right\]\rbrack\)$.

The velocity ($\(\operatorname{mi}\)$/$hr$) of a drone flying in the air is given by $v\(\left\)(t\(\right\))=12+4t^2$ for $0\(\le\) t\(\le\)4$ hours. Let $s\(\left\)(0\(\right\))=0$.

How far has the drone traveled by the time it has reached $48\mathrm{mi}$/$hr$?

A particle moves along the $x$-axis and its acceleration is given by $a\left(t\right)=\cos \pi t$.

Find $v\left(t\right)$ if $v\left(0\right)=0$

A particle travels along the 𝑥-axis and its velocity is the given graph of $v\left(t\right)$. 

Find displacement on $[0,5]$

The velocity ($\(\operatorname{mi}\)$/$hr$) of a drone flying in the air is given by $v\(\left\)(t\(\right\))=12+4t^2$ for $0\(\le\) t\(\le\)4$ hours. Let $s\(\left\)(0\(\right\))=0$.

How far does the drone travel during the first hour?

The velocity ($\mathrm{mi}$/$hr$) of a drone flying in the air is given by $v\left(t\right)=12+4t^2$ for $0\le t\le 4$ hours. Let $s\left(0\right)=0$.

Determine $s\left(t\right)$ for $0\le t\le 4$.

A rock is thrown from a height of $2\mathrm{ft}$ with an initial speed of $25\mathrm{ft}$/$s$. Acceleration resulting from gravity is $-32\mathrm{ft}$/$s^2$.

Find $v\left(t\right)$

A rock is thrown from a height of $2\(\operatorname{ft}\)$ with an initial speed of $25\(\operatorname{ft}\)$/$s$. Acceleration resulting from gravity is $-32\(\operatorname{ft}\)$/$s^2$.

Find $s\left(t\right)$

A particle travels along the 𝑥-axis and its velocity is the given graph of $v\(\left\)(t\(\right\))$. 

Find total distance on $\(\left\[\lbrack\)0,5\(\right\]\rbrack\)$

A particle moves along the $x$-axis and its acceleration is given by $a\(\left\)(t\(\right\))=\(\cos\]\pi\) t$.

Find $s\left(t\right)$ if $s\left(0\right)=1$

Suppose that a particle travels along the $x$-axis and its velocity is given by $v\left(t\right)=2\cos t$ for $0\le t\le 2\pi$.

Find the particle's displacement on $[0,\frac{5\pi }{4}]$