Kinemetics Question - theoretical subsitution

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Homework Statement



This is for personal study and as I am on study leave its a bit hard to contact a teacher.

This is the question; (the circled answer was the teachers answer)

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I'm unsure as to why you would individually square the times before subtracting as the actual time is (t2-t1) substituted into time on the equation, squaring before will result in a larger value (small value for acceleration though). It is possible the teacher merely got it wrong (has been known to happen).

Homework Equations



s=ut + (1/2)at^2

The Attempt at a Solution



Handwritten kinematics working on ruled paper: equations v = u + at, v^2 = u^2 + 2as, s = ut + 1/2 at^2, s = ((u+v)/2)t, with u, v, a, s, t ticked or crossed; below, s = ut + 1/2 a(t2 - t1)^2 with h substituted, leading to a = 2h/(t2 - t1)^2.
 
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Discussion

Your equation,  h = 0(t2-t1) + (1/2)a(t2-t1)2  is in error.

The velocity, u1, at time, t1 is not zero. This velocity is give by:
u1 = u0 + a(t1-t0) = 0 + a t1 = a t1 .​

∴ h = u1(t2-t1) + (1/2)a(t2-t1)2
= (a t1)(t2-t1) + (1/2)a(t2-t1)2​

If you multiply this all out, combine like terms and then factor it, you will find that your teacher did indeed give the correct answer.

BTW: There is an easier way to come up with his answer. I thought it better to use what you started with.