STEP II 2007 question 11 solution

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i)

Finding the distance with Pythagoras' Theorem and then applying the identity twice, we get:

Therefore the minimum distance from the origin is when

The horizontal bearing from O is .

ii) A particle is at its maximum height when the vertical component of its velocity is zero.

When ,

iii) Using the position vector from part i), the distance between O and P is 43.75m. Since we are assuming no forces act on the bullet, it takes = 0.125 seconds to reach P.

It is now necessary to create new velocity and position vectors for the particle with P as an origin.

When t = 0.125,

Ignoring the insignificant value contributed to the total by the change in height, we get:

Clearly this is between 7 and 8. It follows that D approximately equals 3m.

Solution by Dystopia.