How to avoid "what goes up must come down" in motion equations

Question

I am simulating a slot machine spinning and would like to start with an initial velocity and gradually apply a negative acceleration until the total displacement has reached the desired slot. The issue I'm having is: unless I carefully tune the parameters (initial velocity), the slot spins past its target and then reverses direction and finally lands on the correct face.

I used the standard motion equation

displacement = initial_velocity*delta_time + 1/2*acceleration*delta_time^2

to derive the equation for acceleration

acceleration = (2*total_displacement - 2*initial_velocity*delta_time) / delta_time^2

Is there any equation which will tell me what acceleration to use if I want to land directly on the target without reversing? Is acceleration the right concept? I'd greatly appreciate any insights into this issue.

Answer 1

The above equations are correct and shouldn't result in any backwards spinning if used correctly. I was simply running the simulation past the total duration described in the equation (deltaTime), which resulted in a reversal of the velocity direction.

See @Saranga B great answer for even more accurate equations to model a spinning slot, using rotation-based physics.

Answer 2

How much deceleration $a$ to apply depends on the initial velocity $v$. Let D be the displacement, then the correct amount to apply is

$$a = \frac{v^2}{2D}$$

which can be derived from the two equations below,

$$D = vt - \frac{1}{2}at^2$$ $$v=at$$

by eliminating $t$.

Answer 3

I guess its better to think about this problem in terms of angular velocity ($\omega$) and acceleration ($\alpha$). The slot machine stops due to friction, therefore there is constant angular deceleration. You should find the angular velocity that would require to turn the slot machine to a desired target.

$\theta_f$ = $\theta_i + \omega t + \frac{1}{2}\alpha t^2$

So if there are $n$ targets on the rotating dial, then they are $\frac{2\pi}{n}$ radians apart. That is $\theta_f - \theta_i = \frac{2\pi}{n}$. Rearrange the equation to find the desired $\omega$. Hope this helps.