# Solve it kindly?

## A body of mass 3kg is under a force causes a displacement in it ,given by $S = {t}^{2} / 3$(in meters). The work done by the force in 2s is what?

Jul 13, 2017

$W = \frac{8}{3}$ $\text{J}$

#### Explanation:

We're asked to find the work done on a particle given its mass and position equation with time.

I'll solve it without using kinetic energy formulas, but via force and displacement. Take a look at The Useful G.'s response for a different approach.

The equation for the work $W$ done in a particle is the dot product of the force and displacement vectors:

W = vecF • vecs

Since his motion is one-dimensional, we can also write this as

$W = F s$

We can find the object's displacement at time $t = 2$ $\text{s}$ by plugging  in for t in the equation:

$s = \frac{{\left(2\right)}^{2}}{3} = \frac{4}{3}$ $\text{m}$

To find the constant force that acts on the object, we can first find the acceleration as a function of time, by differentiating the position equation twice:

$\frac{{d}^{2}}{{\mathrm{dt}}^{2}} \left[\frac{{t}^{2}}{3}\right] = \frac{2}{3}$

So the acceleration $a$ is constant at $\frac{2}{3}$ ${\text{m/s}}^{2}$.

The force $F$ that acts is

F = ma = (3color(white)(l)"kg")(2/3color(white)(l)"m/s"^2) = color(red)(2 color(red)("N"

Now that we know the force $F$ and displacement $s$, the work done by the force during this time is

W = (color(red)(2)color(white)(l)color(red)("N"))(4/3color(white)(l)"m") = color(blue)(8/3 color(blue)("J"