This section is rated ADVANCED

Now that you understand the basic operations, try some more complex transformations that allow you to express one variable in terms of others. Starting with a simple equation linking speed (*u*), distance (*s*) and time (*t*), it is easy to see that, by using the rules you have already established:

if *u* = *s*/*t*

then *s* = *ut*

and *t* = *s*/*u*

This is for an object traveling at a constant speed. If the object is accelerating at a constant rate, its speed increases with time. If the initial speed is still denoted by *u* and its speed at a time *t* is represented by *v*, then the constant acceleration (*a*) is given by:

*a* = (*v - u*)/*t*

If you know the acceleration, you can transform this equation to find the value of *v* at a particular time, *t*:

*a* = (*v - u*)/*t*

*at* = *v - u*

*v* = *u + at*

From this relationship, you can find the distance traveled (*s*) in a given time (*t*). At the start of this section, you saw that distance is speed multiplied by time. When an object is under constant acceleration, the average speed at a time *t* is half of the sum of the initial speed (*u*) and the speed (*v*) at time *t*. So you can calculate distance traveled as:

*s* = 0.5(*u + v*)*t*

If you don't know the value of *v* but do know the value of *a*, you can replace the value of *v* in the last equation:

*v* = *u + at*

*s* = 0.5(*u + v*)*t*

*s* = 0.5(*u + u + at*)*t*

*s* = *ut* + 0.5 *at*^{2}

(Obviously, if the object is accelerating from rest, *u* is zero, and the equation becomes: *s* = 0.5 *at*^{2})

Alternatively, you can rearrange the same equation to find the value for *v* when you know *u*, *a* and *s* but not *t*. Time (*t*) can be expressed in terms of velocity and acceleration:

*v = u + at*

*at = u - v*

*t* = (*v - u*)/*a*

You can substitute this expression into the earlier equation:

*s = ut* + 0.5 *at*^{2}

*s = u* (*v -u*)/*a* + 0.5 *a* [(*v - u*)/*a*]^{2}

This looks quite daunting. It helps to work out the most complex expression separately. This is represented by a ‘stand-in’ variable, shown here as *A*:

*s = u* (*v - u*)/*a + A*

*A* = 0.5 *a* [(*v - u*)/*a*]^{2}

*A* = 0.5 *a* (*v - u*)^{2}/*a*^{2}

The expression (*v-u*)^{2} means (*v-u*) multiplied by (*v-u*) so that *A* now becomes:

*A* = 0.5 *a* (*v*^{2} - 2*uv + u*^{2})/*a*^{2}

*A* = (0.5*v*^{2}/*a*) - (*uv*/*a*) + (0.5*u*^{2}/*a*)

Having dealt with the complex expression, it can be plugged back into the equation:

*s = u* (*v - u*)/*a + A*

*s = u *(*v -u*)/*a* + 0.5*v*^{2}/*a - uv*/*a* + 0.5*u*^{2}/*a*

*s = uv*/*a - u*^{2}/*a* + 0.5*v*^{2}/*a - uv*/*a* + 0.5*u*^{2}/*a*

*as = uv - u*^{2} + 0.5*v*^{2} - *uv* + 0.5*u*^{2}

*as* = 0.5*v*^{2} - 0.5*u*^{2}

*v*^{2} = *u*^{2} + 2*as*

This section has shown how the simple transforms presented in this document can be used to rearrange equations to derive new relationships between variables, to meet different needs. In this example, the main equations of motion under constant acceleration have been derived from simple expressions relating speed, distance and time. If you understood the much simpler equations earlier in this document, you should have been able to follow the different transformations in this section.