At the top of the hill the skier has purely potential energy. At the bottom, she has purely kinetic energy. 

We can solve by understanding the conservation of energy. The skier's energy at the top of the hill will be equal to her energy at the bottom of the hill.

$\displaystyle PE_{top}=KE_{bottom}$

Using the equations for potential and kinetic energy, we can solve for the height of the hill.

$\displaystyle mgh=\frac{1}{2}mv^2$

The masses cancel, and we can plug in our final velocity and gravitational acceleration.

$\displaystyle gh=\frac{1}{2}v^2$

$\displaystyle (-9.8\frac{m}{s^2})h=\frac{1}{2}(12\frac{m}{s})^2$

$\displaystyle (-9.8\frac{m}{s^2})h=\frac{1}{2}144\frac{m^2}{s^2}$

$\displaystyle (-9.8\frac{m}{s^2})h=72\frac{m^2}{s^2}$

$\displaystyle h=\frac{72\frac{m^2}{s^2}}{-9.8\frac{m}{s^2}}$

$\displaystyle h=-7.35m$

This formula solves for the change in height. The negative sign implies she travelled in a downward direction. Because the question is asking how tall the hill is, we use an absolute value.

We can use conservation of energy to solve. The potential energy when the astronaut is holding the rock will be equal to the kinetic energy just before it strikes the surface.

$\displaystyle PE=KE$

$\displaystyle mgh=\frac{1}{2}mv^2$

Now, we need to solve for $\displaystyle \small g$, the gravity on the new planet. The masses will cancel out.

$\displaystyle gh=\frac{1}{2}v^2$

Plug in the given values and solve.

$\displaystyle g(-10m)=\frac{1}{2}(3\frac{m}{s})^2$

$\displaystyle g(-10m)=\frac{1}{2}(9\frac{m^2}{s^2})$

$\displaystyle g(-10m)=(4.5\frac{m^2}{s^2})$

$\displaystyle g=\frac{4.5\frac{m^2}{s^2}}{-10m}$

$\displaystyle g=-0.45\frac{m}{s^2}$

The equation for potential energy is $\displaystyle PE=mgh$. We are given the mass of the ball, the height of the table, and the acceleration of gravity in the question. The distance the ball travels is in the downward direction, making it negative.

Plug in the values, and solve for the potential energy.

$\displaystyle PE=(0.5kg)(-9.8\frac{m}{s^2})(-1.5m)$

$\displaystyle PE=7.35J$

The units for energy are Joules.

Potential energy can be found using the equation $\displaystyle PE=mgh$. For the pendulum, the height is going to be the length of the string.

Remember, your height is your change in distance. In this case the ball will go down $\displaystyle 1.2m$, so the height will be negative since the ball travels downward.

$\displaystyle PE= (2.03kg)(-9.8\frac{m}{s^2})(-1.2m)$

$\displaystyle PE=23.87J$

The units for energy are Joules.

Use the conservation of energy equation to solve for the potential energy at the top of the hill.

$\displaystyle PE_{top}=KE_{bottom}$

$\displaystyle mgh=\frac{1}{2}mv^2$

Plug in the values given to you and solve for the final height.

$\displaystyle (0.5kg)(9.8\frac{m}{s^2})(h)=\frac{1}{2}(0.5kg)(3.12\frac{m}{s})^2$

$\displaystyle 4.9\frac{kg\cdot m}{s^2}h=2.4336\frac{kg\cdot m^2}{s^2}$

$\displaystyle h=\frac{2.4336\frac{kg\cdot m^2}{s^2}}{4.9\frac{kg\cdot m}{s^2}}$

$\displaystyle h=0.497m\approx0.5m$

The maximum height will be when the ball has only potential energy, and no kinetic energy. Initially, the ball has only kinetic energy and no potential energy. We can set these values equal to each other due to conservation of energy.

$\displaystyle PE_{top}=KE_{bottom}$

$\displaystyle mgh=\frac{1}{2}mv^2$

The masses will cancel out.

$\displaystyle gh=\frac{1}{2}v^2$

Plug in the values that were given and solve for the height.

$\displaystyle (9.8\frac{m}{s^2})h=\frac{1}{2}(3.5\frac{m}{s})^2$

$\displaystyle (9.8\frac{m}{s^2})h=\frac{1}{2}(12.25\frac{m^2}{s^2})$

$\displaystyle (9.8\frac{m}{s^2})h=6.125\frac{m^2}{s^2}$

$\displaystyle h=\frac{6.125\frac{m^2}{s^2}}{9.8\frac{m}{s^2}}$

$\displaystyle h=0.625m$

The only potential energy this ball can have is gravitational potential energy. The formula for gravitational potential energy is $\displaystyle PE=mgh$.

We are given the height and mass of the ball. Using the given values, we can solve for the potential energy.

Keep in mind that the displacement will be negative because the ball is traveling in the downward direction.

$\displaystyle PE=(1.11kg)(-9.8\frac{m}{s^2})(-1.72m)$

$\displaystyle PE=18.71J$

The equation for potential energy is $\displaystyle PE=mgh$.

Since we know the mass, height, and acceleration from gravity, we can simply multiply to find the potential energy.

$\displaystyle PE=(2kg)(-9.8\frac{m}{s^2})(-30m)$

Note that we plugged in $\displaystyle -30m$ for $\displaystyle h$ because the ball will be moving downward; the change in height is negative as the ball drops.

$\displaystyle PE=588J$

The equation for potential energy is $\displaystyle PE=mgh$.

Since we know the mass, height, and acceleration from gravity, we can simply multiply to find the potential energy.

$\displaystyle PE=(4kg)(-9.8\frac{m}{s^2})(-30m)$

Note that we plugged in $\displaystyle -30m$ for $\displaystyle h$ because the ball will be moving downward; the change in height is negative as the ball drops.

$\displaystyle PE=1176J$

The formula for potential energy is $\displaystyle PE=mgh$.

Given the values for the mass, height, and gravity, we can solve using multiplication. Note that the height is negative because the book falls in the downward direction.

$\displaystyle PE=2kg*-9.8\frac{m}{s^2}*-2.3m$

$\displaystyle PE=45.08J$