# The Physics, Astronomy and Mathematics of the Solar System

## Mathematics at the Frontier of Astronomy

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## SECTION 2

### The Mathematical Connections

Mathematics has provided the tools to enable astronomers to quantify their observations. This section will discuss a selection of the concepts and their application to mathematics.

*
*

### Escape Velocity

When a ball is thrown in the air, it rises then it falls back down. If more velocity is applied it would go higher before falling back down. The question is how fast must a ball be thrown so that it does not return that is fall back down. The smallest velocity that will prevent the ball from returning is called escape velocity. Escape velocity is defined as the minimum seed at which a projectile at a planet's surface has to be launched in order to permanently leave the planet.

The task is that we want to give the ball enough kinetic energy so that it will go away to a distance where the Earth's gravitational force is zero therefore using up all the kinetic energy. E
_{
initial
}
+ E
_{
input
}
= E final + E
_{
output
}
gives the equation for the conservation of energy. The E
_{
initial
}
consists of the kinetic and potential energy of the ball when it was released from the earth's surface. After the ball is released there is no energy input or energy output. The ball arrives at infinity with zero kinetic energy and zero potential energy
^{
13
}
.

The equation
*
E
_{
i n i t i a l
}
+E
_{
i n p u t
}
= E
_{
f i n a l
}
+E
_{
o u t p u t
}
*
can be written as

*
½ m v
^{
2
}
*
- (GMm/R)

*+ 0 = 0 + 0*

*
V
_{
e
}
*
= (2Gm/R)

^{ 1/2 }

The relation is applicable for any spherical, symmetrical gravitating body of mass
*
M
*
and radius
*
R
*
. The escape velocity
*
V e
*
depends on the mass and size of the large gravitating body, (e.g) the Earth not the ball. We can therefore calculate the escape velocity for the earth and other planets. We will need to know the mass and the radius of the bodies.