The figure shown below gives the potential energy function $U\left(r\right)$ of a projectile, plotted outward from the surface of a planet of radius $R_{\mathrm{s}}$. If the projectile is launched radially outward from the surface with mechanical energy of $-2.0\times {10}^9 \mathrm{J}$, what are (a) its kinetic energy at radius $r\mathit{=}1.15 R_{\mathrm{s}}$ and (b) its turning point in terms of $R_{\mathrm{s}}$?

In the figure shown below, three $8.00 \mathrm{kg}$ spheres are located at distances $d_1=0.300 \mathrm{m}$ and $d_2=0.400 \mathrm{m}$. What are the (a) magnitude and (b) direction (relative to the positive direction of the $x$-axis) of the net gravitational force on sphere $B$ due to spheres $A$ and $C$?

In the figure shown below, a square of edge length $20.0 \mathrm{cm}$ is formed by four spheres of masses $m_1=5.00 \mathrm{g}\text{,} m_2=3.00 {\mathrm{g}}_1\text{,} m_3=1.00 \mathrm{g}$ and $m_4=5.00 \mathrm{g}$. In unit vector notation, what is the net gravitational force from them on a central sphere with mass $m_5=2.00 \mathrm{g}$?

In deep space, sphere $A$ of mass $20 \mathrm{kg}$ is located at the origin of an $x$-axis and sphere $B$ of mass $10 \mathrm{kg}$ is located on the axis at $x=0.95 \mathrm{m}$. Sphere $B$ is released from rest while sphere $A$ is held at the origin.

$\left(a\right)$ What is the gravitational potential energy of the two sphere system just as $B$ is released?

$\left(b\right)$ What is the kinetic energy of $B$ when it has moved $0.15 \mathrm{m}$ towards $A$?

In $1610$, Galileo used his telescope to discover four moons around Jupiter, with these mean orbital radii $a$ and periods $T$:

$\left(a\right)$ Plot $\log \left(a\right)(\mathrm{y} -\mathrm{axis})$ against $\log \left(T\right)(\mathrm{x}-\mathrm{axis})$, and show that you get a straight line.
$\left(b\right)$ Measure the slope of the line and compare it with the value that you expect from Kepler's third law.
$\left(c\right)$ Find the mass of Jupiter from the intercept of this line with the $y$-axis.

Three identical stars of mass $M$ form an equilateral triangle that rotates around the triangle's centre as the stars move in a common circle about that centre. The triangle has edge length $L$. What is the speed of the stars?

Three point particles are fixed in position in an $xy$-plane. Two of them, particle $A$ of mass $6.00 \mathrm{g}$ and particle $B$ of mass $12.0 \mathrm{g}$, are shown in figure, with a separation of $d_{\mathrm{AB}}=0.500 \mathrm{m}$ at angle $\mathrm{\theta }=30°$. Particle $C$ with mass $10.0 \mathrm{g}$ is not shown. The net gravitational force acting on particle $A$ due to particles $B$ and $C$ is $2.77\times {10}^{-14} \mathrm{N}$ at an angle of $-163.8°$ from the positive direction of the $x$-axis. What are $\left(a\right)$ the $x$ coordinate and $\left(b\right)$ the $y$ coordinate of particle $C$?