10. Physics Applications of Integrals

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. .

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.

105–106. {Use of Tech} Races The velocity function and initial position of Runners A and B are given. Analyze the race that results by graphing the position functions of the runners and finding the time and positions (if any) at which they first pass each other.

A : v(t) = sin t; s(0) = 0 B. V(t) = cos t; S(0) = 0

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.

c. At this rate, how long will it take the racer to travel 1/4 mi?

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = cos2t; v(0) = 5; s(0) = 7

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = e^−t; v(0) = 60; s(0) = 40

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = −32; v(0)=50; s(0)=0