The only force acting on the projectile during its flight is the constant downward vertical force caused by gravity of 9.8 N.

Formulas used.

U = Muzzle Velocity (velocity that projectile leaves the gun, measured in meters per second [m/s], in this case Muzzle Velocity is always 400m/s )

a = Acceleration due to gravity, rounded to a constant 9.8 meters /s/s

Uy = Vertical Velocity

Ux = Horizontal Velocity

In order to calculate the flight time of the projectile, i.e. the amount of time the projectile stays in the air, the actual velocity as it leaves the gun must be resolved into a vertical component.

The vertical component, Uy is calculated by using the formula:

Uy = U Sineθ

i.e. the Muzzle Velocity multiplied by the Sine of the Muzzle’s Angle of Elevation. Sine in trigonometric terms refers to the side opposite the known angle (θ).

The horizontal component Ux is calculated in a similar way, this time by using the formula:

Ux = U Cosineθ

i.e. the Muzzle Velocity multiplied by the Cosine of the angle (θ).

Using the vertical component, the formula v = Uy + a t is re-arranged to find the value of t:

(v – Uy)/a = t

This re-arranged version of the formula is used to calculate the time in seconds before hitting terminal velocity, i.e. ceasing acceleration altogether in the vertical component, before the force of gravity pulls it back towards the ground.

Example:

(0 – 346.41) / -9.8 = 35.31

The value of gravity used here is in the negative because the projectile is travelling upward against the force of gravity.

The total flight time is calculated by doubling this value.

## Excel Formulae

The Sine formula is inserted into the spreadsheet e.g. in the field C5 in the form:

=+SIN(B5*PI()/180)

The Cosine formula is inserted e.g. in the field D5 in the form:

=+COS(B5*PI()/180)

From these we get the vertical component formula in the field E5:

=+A5*C5

Which multiplies the Muzzle Velocity, 400, by the value of the Sine of the Angle of Elevation.

The horizontal component is calculated by the formula in the field F5:

=+A5*D5

The flight time of the projectile formula is inserted in the field G5 in the form:

=+(0-E5)/-9.8

The field H5 multiplies this value by 2 to arrive at the Total Flight Time.

=+G5*2

The Range of the projectile is calculated by multiplying the time by the Horizontal Velocity i.e.:

=+H5*F5

The Maximum Height the missile reaches is calculated by the formula: s = (Uyt)-(9.8/2*t*t), this is entered in to J5 in the form:

=+(E5*G5)-(9.8*0.5*G5*G5)

(These formulas are repeated to their corresponding fields all the way down the column.)

Limit of Accuracy.

The degree of accuracy of the spreadsheet and the resulting graphs is limited by several factors.

The curve of the graph could have been smoother if the intervals had been set at less than 10°.

The computer is limited to using radians, rather than true degrees.

Formulas for converting Degrees into Radians.

Radians

SinΠ = 0

SinΠ /2 = 1

Degrees

Sin180° = 0

Sin 90° = 1

2Π Radians = 360°

360/2Π = 57.29 Radians

As acceleration due to gravity is rounded to one decimal place, the answers are only accurate to approx. 2 decimal places.

The Excel spreadsheet is not really accurate enough for scientific purposes, and is better for business applications.