10. Physics Applications of Integrals

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 $0\le t\le 4$.

The velocity ($\operatorname{mi}$/$hr$) of a drone flying in the air is given by $v\left(t\right)=12+4t^2$ for $0\le t\le4$ hours. Let $s\left(0\right)=0$.

How far does the drone travel during the first hour?

The velocity ($\operatorname{mi}$/$hr$) of a drone flying in the air is given by $v\left(t\right)=12+4t^2$ for $0\le t\le4$ hours. Let $s\left(0\right)=0$.

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

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

Suppose that a particle travels along the $x$-axis and its velocity is given by $v\left(t\right)=2\cos t$ for $0\le t\le2\pi$.

Find total distance traveled by the particle on $\left\lbrack0,\frac{5\pi}{4}\right\rbrack$.

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

Find displacement on $[0,5]$

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

Find total distance on $\left\lbrack0,5\right\rbrack$

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 $t=0.1$ $hr$?

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

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

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 $s\left(t\right)$

Displacement from velocity A particle moves along a line with a velocity given by v(t) = 5 sin πt, starting with an initial position s(0) = 0 . Find the displacement of the particle between t = 0 and t = 2 , which is given by s(t) = ∫₀² v(t) dt . Find the distance traveled by the particle during this interval, which is ∫₀² |v(t)| dt .

Velocity to displacement An object travels on the 𝓍-axis with a velocity given by v(t) = 2t + 5, for 0 ≤ t ≤ 4.

(a) How far does the object travel, for 0 ≤ t ≤ 4 ?

Velocity to displacement An object travels on the 𝓍-axis with a velocity given by v(t) = 2t + 5, for 0 ≤ t ≤ 4.

(c) True or false: The object would travel as far as in part (a) if it traveled at its average velocity (a constant), for 0 ≤ t ≤ 4. .