A roller coaster train is at rest on

To solve the problem, we can use the conservation of energy principle which states that the total energy of a system is constant, i.e., energy cannot be created or destroyed, only transformed from one form to another.
At the top of the hill, the roller coaster train has only potential energy which is given by:
$PE=mgh$
where $m$ is the mass of the roller coaster train, $g$ is the acceleration due to gravity, and $h$ is the height of the hill.
In this case, the potential energy is given as 10,000,000 Joules, and the height of the hill is 50 meters. Since we are not given the mass of the roller coaster train, we cannot directly solve for its velocity using the conservation of energy principle.
However, we can use the fact that the roller coaster train is at rest at the top of the hill to find its initial potential energy. At the half-way point down the hill, the roller coaster train will have lost some of its potential energy and gained some kinetic energy, so we can use the conservation of energy principle to find its velocity at this point.
At the half-way point down the hill, the roller coaster train will have traveled a distance of 25 meters (since the height of the hill is 50 meters). At this point, the roller coaster train will have lost half of its potential energy, which is given by:
$P{E}_{1/2}=\frac{1}{2}PE=\frac{1}{2}mgh$
The roller coaster train will have gained an equal amount of kinetic energy, which is given by:
$K{E}_{1/2}=\frac{1}{2}m{v}^2$
where $v$ is the velocity of the roller coaster train at the half-way point down the hill.
Using the conservation of energy principle, we can set the initial potential energy equal to the sum of the final kinetic and potential energies:
$PE=P{E}_{1/2}+K{E}_{1/2}$
Substituting the expressions for $P{E}_{1/2}$ and $K{E}_{1/2}$, we get:
$mgh=\frac{1}{2}mgh+\frac{1}{2}m{v}^2$
Simplifying and solving for $v$, we get:
$v=\sqrt{gh}=\sqrt{9.8\frac{m}{s^2}·25\text{ m}}=\boxed{22.1\text{ m/s}}$
Therefore, the velocity of the roller coaster train at the half-way point down the hill is 22.1 meters per second.