What is the formula for calculating distance from acceleration data?

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To calculate distance from acceleration data, one can use the average speed during each time interval, derived from acceleration readings. For instance, if starting from 0 m/s and reaching 1 m/s in 10 ms, the average speed is 0.5 m/s, leading to a distance of 0.005 m. If acceleration readings drop to zero, the device continues at the last speed, covering 0.01 m every 10 ms. However, sensor noise can significantly affect accuracy, necessitating advanced techniques like the Kalman filter for effective data smoothing. Accurate distance calculation requires careful handling of these noise factors.
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Homework Statement [/b]

I have a device which gives me acceleration at any desired interval. So let's say i set the interval as 10ms and start moving the device from initial velocity 0. I will get readings for acceleration every 10ms. Using this data i want to calculate the distance which the device has travelled. Can anyone give me a formula for this?
 
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In theory it's simple acceleration is speed/time
So if you started at 0m/s and 10ms later were doing 1 m/s then the average speed during that time interval is (1.0 - 0.0)/2 = 0.5m/s and since distance is speed * time you traveled 0.5m/s * 10/1000 s = 0.005m
If you then have a reading of 0 you are still traveling at the 1m/s speed and so every 10ms you move a further 1.0 * 10/1000 = 0.01m

The problem is that in reality the noise from the sensor will be much larger than the acceleration so you need all sorts of complex averaging to get a sensible answer - look up "Kalman filter"
 
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Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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