# Question #b0468

Mar 30, 2017

The answer is $A + 4 J$

#### Explanation:

The work done is equal to the change in kinetic energy.

$W = F \cdot d$

Here, we have

$W = {\int}_{2}^{1} F \left(x\right) \mathrm{dx}$

$= {\int}_{2}^{1} \left(2 - 4 x\right) \mathrm{dx}$

$= {\left[2 x - \frac{4}{2} {x}^{2}\right]}_{2}^{1}$

$= \left(2 \cdot 1 - 2\right) - \left(4 - 8\right)$

$= 0 - 4 + 8 = 4$

The work done is $= 4 J$

Mar 30, 2017

See below

#### Explanation:

Only to add variety in the types of response this question attracts.

From Newton's 2nd Law:

$F = m \ddot{x} \implies \textcolor{red}{2 - 4 x = m \dot{x} \frac{d \dot{x}}{\mathrm{dx}}}$

We separate and solve the DE:

${\int}_{2}^{1} \setminus 2 - 4 x \setminus \mathrm{dx} = m {\int}_{{v}_{1}}^{{v}_{2}} \setminus \dot{x} \setminus d \dot{x}$

$\left[2 x - 2 {x}^{2}\right] {\setminus}_{2}^{1} = \frac{1}{2} m \left({v}_{2}^{2} - {v}_{1}^{2}\right) \textcolor{b l u e}{= \Delta T}$

$\implies \Delta T = 4 J$