Urgent help on a simple instantaneous power problem

[Blkmage]

Homework Statement

The position of a particle on the x-axis is given by x = 2t^2, due to an applied force of F = e^(2t) parallel to the x axis.

a. Write an expression for the power supplied by this force as a function of time.
b. Compute the instantaneous power when the particle is at x = 8.0

Homework Equations

P = dW/dt
P = Fv

The Attempt at a Solution

So I've talked to some of my classmates about this and I did P = dW/dt while some of them did P = Fv. I'm having trouble comprehending which one to use and why the two methods give different answers.

I said W = Fd = 2t^2*e^(2t) and differentiated that using the product rule, resulting in:

P(t) = 4te^(2t) (t + 1)

Meanwhile, using P = Fv, I just get 4te^(2t). Can someone explain why this is? Part b won't be hard for me once I figure out which way is correct in part a

 

[berkeman]

Homework Statement

The position of a particle on the x-axis is given by x = 2t^2, due to an applied force of F = e^(2t) parallel to the x axis.

a. Write an expression for the power supplied by this force as a function of time.
b. Compute the instantaneous power when the particle is at x = 8.0

Homework Equations

P = dW/dt
P = Fv

The Attempt at a Solution

So I've talked to some of my classmates about this and I did P = dW/dt while some of them did P = Fv. I'm having trouble comprehending which one to use and why the two methods give different answers.

I said W = Fd = 2t^2*e^(2t) and differentiated that using the product rule, resulting in:

P(t) = 4te^(2t) (t + 1)

Meanwhile, using P = Fv, I just get 4te^(2t). Can someone explain why this is? Part b won't be hard for me once I figure out which way is correct in part a

When power is changing over time, I think P=dW/dt would be harder to work with. The formulation P(t)=F(t)*v(t) would seem to lend itself better to calculating instaneous power...

 

[kerimek]

.


As a scientist, it is important to understand the concepts behind the equations rather than just blindly applying them. Both methods can be used to find the instantaneous power, but they are derived from different principles and thus give slightly different answers.

The first method, P = dW/dt, is based on the definition of power as the rate of change of work. In this case, you correctly calculated the work done by the force as W = 2t^2*e^(2t). However, when you differentiate this with respect to time, you also need to use the chain rule to account for the fact that t is a function of time. This leads to the extra term of (t+1) in your final answer.

The second method, P = Fv, is based on the definition of power as the product of force and velocity. In this case, you need to use the expression for velocity, v = dx/dt, to calculate the power. When you substitute this into the equation, you get P = Fv = F(dx/dt) = 4te^(2t), which is the answer you got using this method.

Both methods are valid and can be used to calculate the instantaneous power. It is important to understand the underlying principles and equations to avoid confusion and ensure accuracy in your calculations. In this case, the difference in the answers is due to the fact that the two methods are based on different definitions of power.