Calculus of a Single Variable

Deceleration A car traveling at 45 miles per hour is brought to a stop, at constant deceleration, 132 feet from where the brakes are applied.

(a) How far has the car moved when its speed has been reduced to 30 miles per hour?

(b) How far has the car moved when its speed has been reduced to 15 miles per hour?

(c) Draw the real number line from 0 to $132 .$ Plot the points found in parts (a) and (b). What can you conclude?

Calculus of a Single Variable

A particle, initially at rest, moves along the $x$ -axis such thatits acceleration at time $t>0$ is given by $a(t)=\cos t .$ At time $t=0,$ its position is $x=3$

(a) Find the velocity and position functions for the particle.

(b) Find the values of $t$ for which the particle is at rest.

Calculus of a Single Variable

Escape Velocity The minimum velocity required for an object to escape Earth's gravitational pull is obtained from the solution of the equation

$\int v d v=-G M \int \frac{1}{y^{2}} d y$

where $v$ is the velocity of the object projected from Earth, $y$ is the distance from the center of Earth, $G$ is the gravitational constant, and $M$ is the mass of Earth. Show that $v$ and $y$ are related by the equation

$v^{2}=v_{0}^{2}+2 G M\left(\frac{1}{y}-\frac{1}{R}\right)$

where $v_{0}$ is the initial velocity of the object and $R$ is the radius of Earth.