What is the displacement of the particle

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The discussion focuses on calculating the displacement of a particle given its velocity function, v = (5.5 m/s^4)t^3 - (5.4 m/s). The correct approach involves integrating the velocity function to find the position function, which is the anti-derivative of velocity. The limits of integration should be set from t=3.5s to t=9.0s, and the constant of integration cancels out when evaluating the definite integral. One participant initially miscalculated the integral, leading to an unexpectedly large displacement value. After clarification, they acknowledged the mistake and expressed gratitude for the guidance.
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Here's my problem, The velocity of a particle is given by
v = (5.5 m/s^4)t^3 - (5.4 m/s), what is the displacement of the particle during the interval t=3.5s to t=9.0s? I don't even know if I started it off right. I recalled seeing something similar to this in calc, so I tried integrating the formula given using 3.5 as 'a' and 9.0 as 'b'. I ended up getting 80440.5 as my answer, but it seems a bit large. I just need to know how to approach the problem. Thanks
 
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"a" and "b"? I don't understand what those are! Oh, you mean the limits of integration. That's exactly what you should do.

What you need to know is that velocity is the derivative of position so position is the anti-derivative of velocity- that is, the integral.

Since v= dx/dt= (5.5)t^3 - 5.4, x= (5.5/4)t^4- 5.4t+ x0. (x0 is the "unknown constant" and would be the position of the particle at t=0). Evaluate at t= 9.0s and t= 3.9s and subtract. (The x0 in each term will, of course, cancel).

Why does your answer seem "a bit large"? With the t^3 term the speed is going to increase rapidly!
 
Hey thanks for the help. I figured out what I did wrong. I integrated incorrrectly. THANKS!
 
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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