# Guide to Game Development/Theory/Physical motion/SUVATs

## Basics

SUVAT is a collection of all of the letters used for variables in basic motion equations.

u - The initial velocity (the velocity of the object when it's at its starting position). Unit: ${\displaystyle m/s}$  or ${\displaystyle ms^{-1}}$
v - The final velocity (the velocity of the object when it's at its ending position). Unit: ${\displaystyle m/s}$  or ${\displaystyle ms^{-1}}$
a - The constant acceleration acting on the object. Unit: ${\displaystyle m/s/s}$  or ${\displaystyle ms^{-2}}$
t - The time taken to get from the initial position to get to the final position. Unit ${\displaystyle s}$  or ${\displaystyle sec}$
s - The displacement of the distance travelled (note: displacement is different from distance travelled, e.g. if you were to go 100m in one direction and then 40m back in that same direction, the distance will be 140m, but the displacement will be 60m). Unit ${\displaystyle m}$  or ${\displaystyle meters}$

## The equations you'll need

These are the equations you need to know:

${\displaystyle {\overline {v}}={\frac {\Delta s}{\Delta t}}}$

${\displaystyle {\overline {a}}={\frac {\Delta v}{\Delta t}}}$

${\displaystyle v=u+at}$

${\displaystyle s=ut+{\frac {at^{2}}{2}}}$

${\displaystyle s=vt-{\frac {at^{2}}{2}}}$

${\displaystyle v^{2}=u^{2}+2as}$

${\displaystyle s={\frac {u+v}{2}}t}$

If you were to plot a distance time graph, the gradient would be velocity. If the graph is curved, it shows that there is a change in velocity and the object is accelerating

If you were to plot a velocity time graph, then the area under the graph will represent the displacement, while the gradient of the line represents acceleration. A straight line represents constant acceleration whilst a curved line shows acceleration is changing

If you were to plot an acceleration time graph, then the area under the curve will represent the velocity.

 Example question and answer A car is in a drag race, the race is 1000m (1km) and the driver accelerates with a constant acceleration and finishes the race in 40sec. 1. What's the acceleration of the car? 2. What's the final velocity of the car when it finishes? Answer: ${\displaystyle u=0,s=1000m,t=40sec,a=?,v=?}$  1. Finding ${\displaystyle a}$ : using: ${\displaystyle s=ut+{\frac {at^{2}}{2}}}$  ${\displaystyle 1000=0*40+{\frac {1}{2}}a*40^{2}}$  ${\displaystyle 1000=800a}$  ${\displaystyle a={\frac {1000}{800}}=1.25ms^{-2}}$  2. Finding ${\displaystyle v}$ : using: ${\displaystyle v=u+at}$  ${\displaystyle v=0+1.25*40}$  ${\displaystyle v=50ms^{-1}}$

## Vector quantities

What's a vector? In one sentence, a vector is a quantity with a magnitude and a direction.

u, v, a, s - These can all be represented as vector quantities
t - This is still just a normal variable

Note, that if you want to find either the displacement, speed or acceleration as a number, then you'll need to find the magnitude of the vector, follow these respectfully: ${\displaystyle |s|,|v|,|a|}$