One person drops a ball from the top of a building while another person at the bottom observes its motion. Will these two people agree (a) on the value of the gravitational potential energy of the ball–Earth system? (b) On the change in potential energy? (c) On the kinetic energy of the ball at some point in its motion?

Describe how the kinetic energy and elastic potential energy are lost in a bouncing rubber ball. Describe what happens to the motion of the ball.

If you run down some stairs and stop, what happens to your kinetic energy and your initial gravitational potential energy?

A proton is accelerated through a potential difference of $4.5 \times 10^{6}$ V. How much kinetic energy has the proton acquired?

When a spring is stretched by $5 \mathrm{~cm}$, its elastic potential energy is $1 \mathrm{~J}$. What will its elastic potential energy be if it is compressed by $10 \mathrm{~cm}$?
A. $-4 \mathrm{~J}$
B. $-2 \mathrm{~J}$
C. $2 \mathrm{~J}$
D. $4 \mathrm{~J}$

$(a)$ Derive an expression for the potential energy of an object subject to a force $F_x=a x-b x^3$, where $a=5 \mathrm{~N} / \mathrm{m}$ and $b=2 \mathrm{~N} / \mathrm{m}^3$, taking $U=0$ at $x=0$. $(b)$ Graph the potential energy curve for $x>0$ and use it to find the turning points for an object whose total energy is $-1 \mathrm{~J}$.

A 2 kg ball is released from rest at the top of a track and reaches a speed of 10 m/s at the bottom. a. How much kinetic energy does the ball have? b. How much potential energy did it have at the top of the hill (assuming no energy was lost)? c. What was the height of the hill?

Summation

a. is caused by combining two or more graded potentials.

b. occurs at the trigger zone of the postsynaptic neuron.

c. results in an action potential if it reaches the threshold potential.

d. can occur when two action potentials arrive in close succession at a single presynaptic terminal.

e. All of these are correct.

Use the WKB approximation to estimate the transmission coefficient of a particle of mass $m$ and energy $E$ moving in the following potential barrier:

$$\begin{align*} V(x)= \begin{cases}V_0(x / a+1), & -a<x<0 \\ V_0(1-x / a), & 0<x<a \\ 0, & \text { elsewhere }\end{cases} \end{align*}$$

with $0<E<V_0$; sketch this potential.

A tank of water containing $25 \overline{0} 0 \mathrm{~L}$ of water is stored on the roof a building. (a) Calculate its potential energy with respect to the floor, which is $12.0 \mathrm{~m}$ below the roof. (b) Calculate its potential energy with respect to the basement, which is $4.0 \mathrm{~m}$ below the first floor.

Show that the probability density for the ground-state solution of the one-dimensional Coulomb potential energy has its maximum at $x=a_{0}$.

A parallel-plate capacitor is made of two circular plates, each of which has a diameter of $2.50 \times 10^{−3}$ m. The plates of the capacitor are separated by a space of $1.40 \times 10^{−4}$ m. What is the electrical potential energy stored in the capacitor when fully charged by the potential difference of 0.12 V?

A uniform electric field of magnitude 250 V/m is directed in the positive x direction. A +12.0-$\mu \mathrm { C }$ charge moves from the origin to the point (x, y) = (20.0 cm, 50.0 cm). (a) What is the change in the potential energy of the charge–field system? (b) Through what potential difference does the charge move?

Two point charges $q_1=$ $+2.40 \mathrm{nC}$ and $q_2=-6.50 \mathrm{nC}$ are $0.100 \mathrm{~m}$ apart. Point $A$ is midway between them; point $B$ is $0.080 \mathrm{~m}$ from $q_1$ and $0.060 \mathrm{~m}$ from $q_2$. Take the electric potential to be zero at infinity. Find (a) the potential at point $A$ :