Finding Rectangular Coordinates

AI Thread Summary
To find the rectangular coordinates of the particle's position, velocity, and acceleration at t = 2.5 s, the position vector is given as ~r = (5t^3, 3t - 6t^4). The velocity can be derived by differentiating the position vector with respect to time, resulting in the components of velocity as the first derivatives of the position functions. Similarly, acceleration is obtained by differentiating the velocity components, yielding the second derivatives of the position functions. Understanding the relationships between distance, velocity, and acceleration is crucial, as velocity is the first derivative of position, and acceleration is the first derivative of velocity. The discussion emphasizes the need to derive these functions for accurate calculations.
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Homework Statement



A particle moves with position as a function of time in seconds given by the vector ~r = (5t^3, 3t − 6t^4)m.

What are the rectangular coordinates of the particle’s position at t = 2.5 s?
What are the rectangular coordinates of the particle’s velocity at t = 2.5 s?
What are the rectangular coordinates of the particle’s acceleration at t = 2.5 s?

Homework Equations





The Attempt at a Solution



I'm a little bit lost? I know that the Vector Components is rx=(5t^3) and ry=6t^4 is that right, so I know I plug in numbers to find the vectors but I don't know which ones?
 
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What's the relationship between distance, velocity and acceleration functions?
 
I'm not sure what that means but it says something like this:

Derive expressions for the velocity and the acceleration of the particle as function of time in
Cartesian coordinates.
 
Yes I am asking if you have a function that tells you distance, how do you get velocity function?, how do you get acceleration function?
 
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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