In cases involving only vertical motion we need to change a few assumptions. The forces acting on the character are all in the vertical direction, which results in a slightly different differential equation from the horizontal cases. The four examples I will work out here are each slightly different, but will make use of the same solutions and forms. The results will be directly analagous to those obtained for purely horizontal motion.

Expressing the net vertical force acting on a character accelerating upwards:

**m** **a**
= ( **F**_{thrust} - **m g** ) - **F**_{friction}

To simplify the mathematics we will define a new maximum speed. When a character is flying upwards he will have a different (smaller) maximum velocity than when flying horizontally. This upwards maximum velocity can be expressed using:

**F**_{thrust} - **m g** = **c**_{friction}**
v**_{max-up}^{2}

**v**_{max-up}^{2} = ( **F**_{thrust} - **m g** ) / **c**_{friction}

This will simplify the differential equation, and the solution will be similar to that of horizontal acceleration:

**m** ( d**v**
/ d**t** ) = **c**_{friction}
(** v**_{max-up}^{2} - **v**^{2} )

**t**_{accel} = **m** / ( 2 **c**_{friction}
**v**_{max-up} ) * *ln* { ( **v**_{max-up} +** v** ) /
( **v**_{max-up} -** v**
) }

If we make the definition: **v** = **N** **v**_{max-up}
we obtain the result:

**t**_{accel} = **m** / ( 2 **c**_{friction}
**v**_{max-up} ) * *ln* { ( 1 +** ****N** ) / ( 1 -** ****N**
) }

**t**_{accel} = **m**
**f**_{N} / ( **c**_{friction}
**v**_{max-up} )

The character is travelling upwards and wishes to come to a complete stop. He now must point his thrust downwards, giving us the differential equation:

**m** ( d**v**
/ d**t** ) = ( **F**_{thrust}
+ **m g** ) + **F**_{friction}

Again, to simplify the mathematics we will define a new maximum speed. A character's maximum possible downward velocity will be greater than his maximum horizontal velocity because gravity is pulling him downwards. This maximum velocity is determined using:

**F**_{thrust} + **m g** = **c**_{friction}**
v**_{max-down}^{2}

**v**_{max-down}^{2} = ( **F**_{thrust} + **m g** ) / **c**_{friction}

With the definition: **v** = **N**
**v**_{max-down} and substitution of our standard
frictional coefficients we obtain:

**t**_{decel} = **m** / ( **c**_{friction}
**v**_{max-down} ) * *tan*^{-1} ( **N** )

**t**_{decel} = **m**
**s**_{N} / ( **c**_{friction}
**v**_{max-down} )

We obtain the differential equation and the solution is obtained:

**m** ( d**v**
/ d**t** ) = ( **F**_{thrust}
+ **m g** ) - **F**_{friction}

**t**_{accel} = **m** **f**_{N}
/ (** c**_{friction} **v**_{max-down}
)

We obtain the differential equation and the solution is obtained:

**m** ( d**v**
/ d**t** ) = ( **F**_{thrust}
- **m g** ) + **F**_{friction}

**t**_{decel} = **m** **s**_{N}
/ ( **c**_{friction} **v**_{max-up}
)

The four cases of acceleration and deceleration solved above can be generalized:

**v** = **N**
**v**_{max-direction}

**t**_{accel} = **m** **f**_{N}
/ ( **c**_{friction} **v**_{max-direction}
)

**t**_{decel} = **m**
**s**_{N} / ( **c**_{friction}
**v**_{max-direction} )

To decide which value of **v**_{max} is needed
you must look at the direction of your character's thrust. If your character is pushing
upwards (accelerating upwards or decelerating while travelling downwards) you need to use **v**_{max-up}. If your character is pushing downwards
(accelerating downwards or decelerating while travelling upwards) you need to use **v**_{max-down}. This generalization also applies to
the case of horizontal acceleration and deceleration, where you use the standard value of **v**_{max}. The benefits of making this generalization
should be clear: the values of **f**_{N}
and **s**_{N} and the forms of the
equations used are the same regardless of the direction in which you're travelling! If a
player knows his character's maximum upwards, horizontal, and downwards velocities he can
use a few sample values of **f**_{N} and
**s**_{N} to solve for several
acceleration and deceleration times without depending on excessive tables and
calculations.