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Making t the subject of an equation containing t2
By Trevor Phillips on January 11, 1999:

s = ut + 1/2at2

How can I make t the subject of this equation?

thanks
TP

By Gordon Lee (gtyl2) on January 11, 1999:

Trevor,

Just complete the square!

s = a/2 (2ut/a + t2)
= a/2 ( (t + u/a)2 - (u/a)2 )

Now, I am sure you can do the rest

Gordon

By The Editor:

If you haven't learnt how to complete the square, read on:

Suppose I square the bracket (t+k)
(t+k)2 = (t+k)(t+k) = t2 + 2kt + k2

Now we're going to try and work backwards to make the ut + 1/2at2 into a squared bracket.

First of all, we'll make that 1/2at2 into t2, by multiplying through by 2 and dividing by a:
2s/a = 2ut/a + t2 ---(1)

Now compare the term with t in it to the 2kt we had above. 2k must be the same as 2u/a, so k must be u/a. So let's suppose we square (t+u/a):
(t+u/a)(t+u/a) = t2 + 2ut/a + u2/a2

That's almost the same as we had on the right-hand-side at (1).
So (t+u/a)2 = 2s/a + u2/a2
Now you need to square root both sides, and if you could do all that, the next bit really is easy.


Rearranging equations like this, where you have t and t2, is significantly more difficult than when there is no squared term, and completing the square is the way to make it so that t only appears once.

A useful way to check your algebra is:

  • Make up values for u, t, a
  • Use the original equation to work out s
  • Use the values of u, a and s and your new equation to work out t
  • Check t turns out to be the value you started with