Power, Work, Kinetic Energy Problem

AI Thread Summary
The discussion revolves around solving a physics problem involving a 4.60 kg particle moving along the x-axis, where its position is defined by the equation x = t + 1.8t³. Participants are tasked with finding the kinetic energy, acceleration, force, power, and work done on the particle over a specified time interval. Key calculations include differentiating the position function to obtain velocity and acceleration, and using these to derive kinetic energy and power. There is some confusion about the need for initial conditions and how to apply the work-energy theorem effectively. The conversation highlights the interconnectedness of these concepts in solving the problem accurately.
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Homework Statement



A 4.60 kg particle moves along the x axis. Its position varies with time according to x = t + 1.8t3, where x is in meters and t is in seconds.

(a) Find the kinetic energy at any time t. (Accurately round any coefficient to exactly two decimal places. Use t as necessary.)

(b) Find the acceleration of the particle and the force acting on it at time t. (Accurately round any coefficient to exactly two decimal places. Use t as necessary.)

(c) Find the power being delivered to the particle at time t. (Accurately round any coefficient to exactly two decimal places. Use t as necessary.)

(d) Find the work done on the particle in the interval t = 0 to t = 2.00 s.


Homework Equations



Im pretty that equations arent normal variable equations arent used here because I am asked for expressions, not just a numerical value.

The Attempt at a Solution



I've tried to fiddle around a little bit, but I've basically just made a mess of wrong answers... can someone please help me out here. I am very confused even on where to start on this question.
 
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yb1013 said:

Homework Statement



A 4.60 kg particle moves along the x axis. Its position varies with time according to x = t + 1.8t3, where x is in meters and t is in seconds.

(a) Find the kinetic energy at any time t. (Accurately round any coefficient to exactly two decimal places. Use t as necessary.)

(b) Find the acceleration of the particle and the force acting on it at time t. (Accurately round any coefficient to exactly two decimal places. Use t as necessary.)

(c) Find the power being delivered to the particle at time t. (Accurately round any coefficient to exactly two decimal places. Use t as necessary.)

(d) Find the work done on the particle in the interval t = 0 to t = 2.00 s.


Homework Equations



I'm pretty that equations aren't normal variable equations aren't used here because I'm asked for expressions, not just a numerical value.

The Attempt at a Solution



I've tried to fiddle around a little bit, but I've basically just made a mess of wrong answers... can someone please help me out here. I'm very confused even on where to start on this question.

I'm going to guess that's x(t) = t + 1.8t3... Well it's velocity as a function of time is v(t) = 1 + 5.4t2 and acceleration is a(t) = 10.8t both by differentiation.

Thus kinetic energy is given by KE = .5mv2.

Acceleration is given by what is above (the double derivative of x(t). F = ma.

What does power mean? Work from that definition.

Try the work/kinetic energy theorem for the the last part but you need initial conditions (namely initial velocity).
 
hmmm, I understood your first two explanations, but still kinda puzzled about the last 2... can you use Power = W/t
 
The last two are tied together.

If you do P = W/t then you need to calculate work.

W = \Delta KE = KE_{final} - KE_{initial}. But to find KE you need a velocity which for some reason I thought needed a given value for t=0 but it's just v(0) = 1+5.4t2 = 1 m/s.

Good?
 
ok i think i got it, thank you
 
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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