Solving for acceleration of a piston

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AI Thread Summary
The discussion focuses on calculating the average acceleration of a piston with a displacement of 10 cm and a frequency of 50 cycles per second. The formula d = .5a * t² is used to derive the acceleration, with the need to first compute average velocity and the time taken to travel 5 cm. The user calculates that one cycle takes 20 milliseconds, leading to a velocity of 10 m/s and a time of 5 milliseconds for the piston to reach the midpoint. However, confusion arises regarding the correct distance used in the acceleration formula, as the user mistakenly uses 0.1 m instead of the appropriate distance for the specific time interval. The conversation highlights the importance of accurately defining variables in physics calculations.
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Homework Statement


The piston has a displacement of 10 cm and there are 50 cycles per second. Calculate the average acceleration that occurs between the start/stop point and the middle of the cycle. This value has to do with evenly accelerated movement. This can be described according to the formula d = .5a * t2.
To solve:
-Compute the average velocity using a full cycle.
-Calculate how long the piston needs to travel 5 cm from the top of the cycle to the middle, which will give us the value for t. Then use a = 2 * d / t2 to calculate the acceleration.

Homework Equations


d = .5a * t2
a = 2 * d / t2
v = d / t

The Attempt at a Solution


50 cycles per second = 1 cycle takes 20 milliseconds.
velocity = d / t
10 m/s = 20cm / 50 ms (total cycle distance is 20cm because the displacement is 10cm)
How long the piston takes to travel 5 cm: 5 ms (so t = .005s)
a = 2 * d / t2
a = 2 * .1m / .005s2 = 40 m/s2
But this isn't the right answer! Please help!
 
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Why are you using .1 m for d? d is the distance traveled in time t

AM
 
Thread 'Variable mass system : water sprayed into a moving container'

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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