Find force with mass and variable position

[zrbecker]

Homework Statement

A force acts on a 2.80 kg particle in such a way that the position of the particle as a function of time is given by x = (3.0 m/s)t - (4.0 m/s^2)t^2 + (1.0 m/s^3)t^3. Find the work done by the force during the first 4.0s.

Homework Equations

$$W = \int_{t_i}^{t_f} F(t) dt$$

$$F = ma$$

The Attempt at a Solution

$$W = \int_0^4 F(t) dt = \int_0^4 ma(t) dt = mv(4) - mv(0) = m(x'(4) - x'(0))$$

$$x'(t) = 3 - 8t + 3t^2$$
$$x'(4) = 19$$
$$x'(0) = 3$$

$$W = 2.8 * 16 = 44.8J$$

The book is saying it is 493J

 

[fearTheEcma]

The integral of force with respect to time is not the work done by a force; it is the impulse of that force. The easiest way to convince yourself that there is a problem with your equations is by dimensional analysis. In your final equation, you have 2.8 kg * 16 m/s or 44.8 N*s. A Joule is a N*m.

So the first step is to double check your definition of work.

Otherwise, I think that you have an excellent start on the problem.

 

[zrbecker]

Ah thanks. Yeah my bad it is suppose to be the integral with respect to position.

I see why that doesn't work now.

I found the better way to do this was to use

W = K_f - K_i

and that got me to where I wanted to go. Thanks for the help.