How do you derive the fifth equation of motion ( i.e. SUVAT equations)?

Question

I understand how to derive the following 4 SUVAT equations:

(1) v = u + at

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(2) S = ut + 0.5at^2

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(3) S = 0.5(u + v) x t

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(4) v^2 = u^2 = 2as

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I understand that the 4th equation is obtained by rearranging equation (1) to make 't' the subject and subbing that into equation (3). After, some manipulation you end up at the equation shown above.

However, during my mechanics class I was told that there was an extra SUVAT but that equation isn't necessary. I was wondering if anyone knows what it is and how to derive it.

Answer 1

Solve the first equation for $u$: $$ u = v - at $$ and substitute into the second: $$ s = ut + \tfrac{1}{2}at^2 = (v - at)t + \tfrac{1}{2}at^2 = vt - at^2 + \tfrac{1}{2}at^2 = vt + (-1 + \tfrac{1}{2})at^2 = vt - \tfrac{1}{2}at^2 $$

Answer 2

I suppose that the five equations that you are searching are:

$$ v=at+v_0 $$ $$ s=s_0+v_0t+\frac{1}{2}at^2 $$ $$ s=s_0+\frac{1}{2}(v+v_0)t $$ $$ v^2=v_0^2+2a(s-s_0) $$ $$ s=s_0+vt-\frac{1}{2}at^2 $$

that can be found from the equation of motion for a particle moving on a straight line with constant acceleration.