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STEP III 1990 question 13 solution
From The Student RoomTSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP III 1990 question 13 solution Consider a general point P which makes an angle The forces acting on the particle are its weight, acting downwards with magnitude mg, and the reaction force R acting towards O. Let the radius of sphere be r.
The particle leaves the surface when R = 0.
Since the velocity is tangential to the circle, the resolved velocity vector makes an angle of Therefore, the velocity and position vectors are:
Let the collision occur at some point C. We're trying to find the angle which The horizontal distance to C is Inserting this value of t into the equation for the height:
The change in height is
When you multiply this all out and simplify, you end up with For C to be below O,
Energy is conserved, so Where v = velocity at P, V = velocity at C. This simplifies to Using the velocity vector, we find When this is simplified you end up with the same cubic equation in terms of Solution by Dystopia. |










with the upwards vertical.
; the horizontal velocity is a constant
. Therefore the collision occurs at
.
. Equating the two you get:
; the only real solution to this is
, so when the collision is at the same height as O,
.
.
.
.





