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 $[0,\frac{5\pi }{4}]$

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. .

43. A hot-air balloon is launched from an elevation of 5400 ft above sea level. As it rises, the vertical velocity is computed using a device (called a variometer) that measures the change in atmospheric pressure. The vertical velocities at selected times are shown in the table (with units of ft/min).

c. A polynomial that fits the data reasonably well is:

g(t) = 3.49t³ - 43.21t² + 142.43t - 1.75

Estimate the elevation of the balloon after five minutes using this polynomial.