In a cathode ray tube, electrons initially at rest are accelerated by a uniform electric field of magnitude $4.0 \times 10 ^ { 5 } \mathrm { N } / \mathrm { C }$ during the first 5.0 cm of the tube's length; then they move at essentially constant velocity another 45 cm before hitting the screen. (a) Find the speed of the electrons when they hit the screen. (b) How long does it take them to travel the length of the tube?

Graph each equation in a

$$\left[ - 2 \pi , 2 \pi , \frac { \pi } { 2 } \right]$$

by [-3, 3, 1] viewing rectangle. Then a. Describe the graph using another equation, and b. Verify that the two equations are equivalent.

$$y = \frac { 2 \tan \frac { x } { 2 } } { 1 + \tan ^ { 2 } \frac { x } { 2 } }$$

You are standing on a bathroom scale inside an elevator. Your weight is 140 lb, but the reading of the scale is 120 lb. (a) What is the magnitude and direction of the acceleration of the elevator? (b) Can you tell whether the elevator is speeding up or slowing down?

In lab tests it was found that rats can detect electric fields of about 5.0 kN/C or more. If a point charge of 1.0$\mu C$ is sitting in a maze, how close must the rat come to the charge in order to detect it?

Suppose $A$ and $B$ are sets, and $f$ and $g$ are functions with $f:A\rightarrow B$ and $g:B\rightarrow A$. Prove: If $g\circ f=\text{id}_A$ and $f\circ g=\text{id}_B$, then $f$ is invertible and $g=f^{-1}$. Note: This result is a converse to Proposition 26.9.

(a) Give a counterexample to show that $$ ( A B ) ^ { - 1 } \neqA ^ { - 1 } B ^ { - 1 } $$ in general. (h) Under what conditions on A and B is $$ ( A B ) ^ { - 1 } =A ^ { - 1 } B ^ { - 1 } ? $$ Prove your assertion.

Solve each system for x and y expressing either value in terms of a or b, if necessary. Assume that

$$a \neq 0 \text { and } b \neq 0.$$

For the linear function

$$f ( x ) = m x + b , f ( - 2 ) = 11 \text { and } f ( 3 ) = - 9. Find m and b.$$

The length of a rectangle is 2 inches more than the width w. What expression in terms of w could you use for the length of the rectangle?

Show that a finite abelian group is not cyclic if and only if it contains a subgroup isomorphic to

$$ℤ_p × ℤ_p$$

for some prime p.

Determine the decimal value of each signed binary number in the 2’s complement form: (a) 10011001 (b) 01110100 (c) 10111111

A 1-m^3 tank contains 2.841 kg of steam at 0.6 MPa. Determine the temperature of the steam, using (a) the ideal-gas equation, (b) the van der Waals equation, and (c) the steam tables. Answers: (a) 457.6 K, (b) 465.9 K, (c) 473 K

Some forms of cancer can be treated using proton therapy in which proton beams are accelerated to high energies, then directed to collide into a tumor, killing the malignant cells. Suppose a proton accelerator is 4.0 m long and must accelerate protons from rest to a speed of $1.0 \times 10 ^ {7} \ \text{m}/\text{s}$. Ignore any relativistic effects and determine the magnitude of the average electric field that could accelerate these protons.

A charge of 63.0 nC is located at a distance of 3.40 cm from a charge of -47.0 nC. What are the x- and y-components of the electric field at a point P that is directly above the 63.0 nC charge at a distance of 1.40 cm? Point P and the two charges are on the vertices of a right triangle.

Two otherwise identical conducting spheres carry charges of +5.0

$$\mu C$$

and -1.0

$$\mu C$$

. They are initially a distance L apart. The distance L is much larger than the radii of the spheres. The spheres are brought together, touched together, and then returned to their original separation L. What is the ratio of the magnitude of the force on either sphere after they are touched to that before they were touched?