STEP I 1990 question 12 solution

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Tension in spring using modulus of elasticity:

Tension in string by resolving vertically:

Tension in string by resolving horizontally:

As required.

Using elastic potential energy = , we get

Using Pythagoras, the change in height is

Change in gravitational potential energy =

Energy is conserved, so

When the particle comes to rest, v = 0, so:

Substituting for mg, we get:

Note that in 1990 calculators were allowed, so it is easy to numerically confirm that a possible solution for x is approximately 0.66:

Solution by Dystopia.