$A$ woman pushes a trunk on a railway platform which has a rough surface. She applies a force of $100 \; N$ over a distance of $10 \; m$. Thereafter,she gets progressively tired and her applied force reduces linearly with distance to $50 \; N$. The total distance through which the trunk has been moved is $20 \; m$. Plot the force applied by the woman and the frictional force,which is $50 \; N$ versus displacement. Calculate the work done by the two forces over $20 \; m$.

(N/A) The plot of the applied force is shown in the figure. At $x = 20 \; m$,the applied force $\vec{F} = 50 \; N \neq 0$. We are given that the frictional force $f$ is $|f| = 50 \; N$. It opposes motion and acts in a direction opposite to $F$. It is therefore shown on the negative side of the force axis.
The work done by the woman is:
$W_F = \text{Area of rectangle } ABCD + \text{Area of trapezium } CDEI$
$W_F = (100 \; N \times 10 \; m) + \frac{1}{2} \times (100 \; N + 50 \; N) \times (20 \; m - 10 \; m)$
$W_F = 1000 \; J + \frac{1}{2} \times 150 \; N \times 10 \; m$
$W_F = 1000 \; J + 750 \; J = 1750 \; J$
The work done by the frictional force is:
$W_f = \text{Area of rectangle } AGHI$
$W_f = (-50 \; N) \times 20 \; m$
$W_f = -1000 \; J$
The area on the negative side of the force axis has a negative sign.