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I figured 1a out pretty easily.

For 1b I am a bit stuck. So far i have that in the field: $$ \Delta z = v_0t + (1/2) a_zt $$ $$ F_z = ma_z $$ so $$ a_z = F_z/m $$ We know $$ F_z=\mu_z(\partial(B_z)/\partial(z)) $$ So knowing the time we can find the displacement $$\Delta(z)$$

Out of the field its constant veloctiy so $$\Delta(z)=v_{out}*t$$

What can i do to relate these though to find the minimum gradient.

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closed as off-topic by user10851, Kyle Kanos, ACuriousMind, user36790, John Rennie Mar 28 '16 at 15:06

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The atom is only being accelerated vertically with $F_z/m$ in $z$ direction as it passes through the $1m$ long magnet. Since you know the horizontal velocity, you also know the time $t_1$ for the acceleration.

After passing the magnet the atom moves with a constant velocity both in horizontal and in vertical directions, where the vertical velocity is $$v_z=\frac{F_z}{m} t_1.$$ By writing down four equations of motion in this style and solving them you can obtain the value of $F_z$.

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